Merge pull request #31 from kalmarek/mk/update_to_PG_0.6

Update to PermutationGroups-0.6
bump to v0.8
2024-02-13 12:08:48 +01:00 · 2024-02-13 11:16:04 +01:00 · 2024-02-12 14:04:07 +01:00 · 2024-02-12 12:38:28 +01:00 · 2024-02-12 12:37:33 +01:00 · 2024-02-12 12:36:04 +01:00
37 changed files with 948 additions and 487 deletions
--- a/.JuliaFormatter.toml
+++ b/.JuliaFormatter.toml
@ -0,0 +1 @@
+../.JuliaFormatter.toml
--- a/.github/workflows/CompatHelper.yml
+++ b/.github/workflows/CompatHelper.yml
@ -7,7 +7,11 @@ jobs:
  CompatHelper:
    runs-on: ubuntu-latest
    steps:
-      - uses: JuliaRegistries/compathelper-action@v1
-        with:
-          token: ${{ secrets.GITHUB_TOKEN }}
-          ssh: ${{ secrets.DOCUMENTER_KEY }}
+      - name: Pkg.add("CompatHelper")
+        run: julia -e 'using Pkg; Pkg.add("CompatHelper")'
+      - name: CompatHelper.main()
+        env:
+          GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
+          COMPATHELPER_PRIV: ${{ secrets.DOCUMENTER_KEY }}
+        run: julia -e 'using CompatHelper; CompatHelper.main()'
+
--- a/Project.toml
+++ b/Project.toml
@ -1,24 +1,23 @@
 name = "Groups"
 uuid = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
 authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
-version = "0.7.4"
+version = "0.8"

 [deps]
-Folds = "41a02a25-b8f0-4f67-bc48-60067656b558"
 GroupsCore = "d5909c97-4eac-4ecc-a3dc-fdd0858a4120"
 KnuthBendix = "c2604015-7b3d-4a30-8a26-9074551ec60a"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Logging = "56ddb016-857b-54e1-b83d-db4d58db5568"
 OrderedCollections = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
 PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
+Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"

 [compat]
-Folds = "0.2.7"
-GroupsCore = "0.4"
+GroupsCore = "0.5"
 KnuthBendix = "0.4"
 OrderedCollections = "1"
-PermutationGroups = "0.3"
+PermutationGroups = "0.6"
 StaticArrays = "1"
 julia = "1.6"

--- a/README.md
+++ b/README.md
@ -1,5 +1,5 @@
 # Groups
-[![CI](https://github.com/kalmarek/Groups.jl/actions/workflows/runtests.yml/badge.svg)](https://github.com/kalmarek/Groups.jl/actions/workflows/runtests.yml)
+[![CI](https://github.com/kalmarek/Groups.jl/actions/workflows/ci.yml/badge.svg)](https://github.com/kalmarek/Groups.jl/actions/workflows/ci.yml)
 [![codecov](https://codecov.io/gh/kalmarek/Groups.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/kalmarek/Groups.jl)

 An implementation of finitely-presented groups together with normalization (using Knuth-Bendix procedure).
--- a/src/Groups.jl
+++ b/src/Groups.jl
@ -1,12 +1,9 @@
 module Groups

-import Folds
 import Logging

 using GroupsCore
-import GroupsCore.Random
-
-import OrderedCollections: OrderedSet
+import Random

 import KnuthBendix
 import KnuthBendix: AbstractWord, Alphabet, Word
@ -14,7 +11,14 @@ import KnuthBendix: alphabet, ordering

 export MatrixGroups

-export Alphabet, AutomorphismGroup, FreeGroup, FreeGroup, FPGroup, FPGroupElement, SpecialAutomorphismGroup, Homomorphism
+export Alphabet,
+    AutomorphismGroup,
+    FreeGroup,
+    FreeGroup,
+    FPGroup,
+    FPGroupElement,
+    SpecialAutomorphismGroup,
+    Homomorphism

 export alphabet, evaluate, word, gens

@ -23,6 +27,7 @@ include(joinpath("constructions", "constructions.jl"))
 import .Constructions

 include("types.jl")
+include("rand.jl")
 include("hashing.jl")
 include("normalform.jl")
 include("autgroups.jl")
--- a/src/abelianize.jl
+++ b/src/abelianize.jl
@ -1,7 +1,8 @@
 function _abelianize(
    i::Integer,
    source::AutomorphismGroup{<:FreeGroup},
-    target::MatrixGroups.SpecialLinearGroup{N,T}) where {N,T}
+    target::MatrixGroups.SpecialLinearGroup{N,T},
+) where {N,T}
    n = ngens(object(source))
    @assert n == N
    aut = alphabet(source)[i]
@ -12,7 +13,7 @@ function _abelianize(
        eij = MatrixGroups.ElementaryMatrix{N}(
            aut.j,
            aut.i,
-            ifelse(aut.inv, -one(T), one(T))
+            ifelse(aut.inv, -one(T), one(T)),
        )
        k = alphabet(target)[eij]
        return word_type(target)([k])
@ -24,7 +25,8 @@ end
 function _abelianize(
    i::Integer,
    source::AutomorphismGroup{<:Groups.SurfaceGroup},
-    target::MatrixGroups.SpecialLinearGroup{N,T}) where {N,T}
+    target::MatrixGroups.SpecialLinearGroup{N,T},
+) where {N,T}
    n = ngens(Groups.object(source))
    @assert n == N
    g = alphabet(source)[i].autFn_word
@ -39,7 +41,7 @@ end
 function Groups._abelianize(
    i::Integer,
    source::AutomorphismGroup{<:Groups.SurfaceGroup},
-    target::MatrixGroups.SymplecticGroup{N,T}
+    target::MatrixGroups.SymplecticGroup{N,T},
 ) where {N,T}
    @assert iseven(N)
    As = alphabet(source)
@ -50,10 +52,10 @@ function Groups._abelianize(
        MatrixGroups.SpecialLinearGroup{2genus}(T)
    end

-    ab = Groups.Homomorphism(Groups._abelianize, source, SlN, check=false)
+    ab = Groups.Homomorphism(Groups._abelianize, source, SlN; check = false)

    matrix_spn_map = let S = gens(target)
-        Dict(MatrixGroups.matrix_repr(g) => word(g) for g in union(S, inv.(S)))
+        Dict(MatrixGroups.matrix(g) => word(g) for g in union(S, inv.(S)))
    end

    # renumeration:
@ -63,7 +65,7 @@ function Groups._abelianize(
    p = [reverse(2:2:N); reverse(1:2:N)]

    g = source([i])
-    Mg = MatrixGroups.matrix_repr(ab(g))[p, p]
+    Mg = MatrixGroups.matrix(ab(g))[p, p]

    return matrix_spn_map[Mg]
 end
--- a/src/aut_groups/gersten_relations.jl
+++ b/src/aut_groups/gersten_relations.jl
@ -1,4 +1,4 @@
-function gersten_alphabet(n::Integer; commutative::Bool=true)
+function gersten_alphabet(n::Integer; commutative::Bool = true)
    indexing = [(i, j) for i in 1:n for j in 1:n if i ≠ j]
    S = [ϱ(i, j) for (i, j) in indexing]

@ -40,12 +40,17 @@ function _hexagonal_rule(
    return W(T[A[x], A[inv(y)], A[z]]) => W(T[A[z], A[w^-1], A[x]])
 end

-gersten_relations(n::Integer; commutative) =
-    gersten_relations(Word{UInt8}, n, commutative=commutative)
+function gersten_relations(n::Integer; commutative)
+    return gersten_relations(Word{UInt8}, n; commutative = commutative)
+end

-function gersten_relations(::Type{W}, n::Integer; commutative) where {W<:AbstractWord}
+function gersten_relations(
+    ::Type{W},
+    n::Integer;
+    commutative,
+) where {W<:AbstractWord}
    @assert n > 1 "Gersten relations are defined only for n>1, got n=$n"
-    A = gersten_alphabet(n, commutative=commutative)
+    A = gersten_alphabet(n; commutative = commutative)
    @assert length(A) <= typemax(eltype(W)) "Type $W can not represent words over alphabet with $(length(A)) letters."

    rels = Pair{W,W}[]
@ -74,7 +79,10 @@ function gersten_relations(::Type{W}, n::Integer; commutative) where {W<:Abstrac

    for (i, j, k) in Iterators.product(1:n, 1:n, 1:n)
        if (i ≠ j && k ≠ i && k ≠ j)
-            push!(rels, _pentagonal_rule(W, A, ϱ(i, j)^-1, ϱ(j, k)^-1, ϱ(i, k)^-1))
+            push!(
+                rels,
+                _pentagonal_rule(W, A, ϱ(i, j)^-1, ϱ(j, k)^-1, ϱ(i, k)^-1),
+            )
            push!(rels, _pentagonal_rule(W, A, ϱ(i, j)^-1, ϱ(j, k), ϱ(i, k)))

            commutative && continue
@ -83,7 +91,10 @@ function gersten_relations(::Type{W}, n::Integer; commutative) where {W<:Abstrac
            push!(rels, _pentagonal_rule(W, A, ϱ(i, j), λ(j, k)^-1, ϱ(i, k)))

            # the same as above, but with ϱ ↔ λ:
-            push!(rels, _pentagonal_rule(W, A, λ(i, j)^-1, λ(j, k)^-1, λ(i, k)^-1))
+            push!(
+                rels,
+                _pentagonal_rule(W, A, λ(i, j)^-1, λ(j, k)^-1, λ(i, k)^-1),
+            )
            push!(rels, _pentagonal_rule(W, A, λ(i, j)^-1, λ(j, k), λ(i, k)))

            push!(rels, _pentagonal_rule(W, A, λ(i, j), ϱ(j, k), λ(i, k)^-1))
@ -94,7 +105,10 @@ function gersten_relations(::Type{W}, n::Integer; commutative) where {W<:Abstrac
    if !commutative
        for (i, j) in Iterators.product(1:n, 1:n)
            if i ≠ j
-                push!(rels, _hexagonal_rule(W, A, ϱ(i, j), ϱ(j, i), λ(i, j), λ(j, i)))
+                push!(
+                    rels,
+                    _hexagonal_rule(W, A, ϱ(i, j), ϱ(j, i), λ(i, j), λ(j, i)),
+                )
                w = W([A[ϱ(i, j)], A[ϱ(j, i)^-1], A[λ(i, j)]])
                push!(rels, w^2 => inv(w, A)^2)
            end
--- a/src/aut_groups/mcg.jl
+++ b/src/aut_groups/mcg.jl
@ -17,7 +17,7 @@ function Base.show(io::IO, S::SurfaceGroup)
    end
 end

-function SurfaceGroup(genus::Integer, boundaries::Integer, W=Word{Int16})
+function SurfaceGroup(genus::Integer, boundaries::Integer, W = Word{Int16})
    @assert genus > 1

    # The (confluent) rewriting systems comes from
@ -31,7 +31,15 @@ function SurfaceGroup(genus::Integer, boundaries::Integer, W=Word{Int16})
    ltrs = String[]
    for i in 1:genus
        subscript = join('₀' + d for d in reverse(digits(i)))
-        append!(ltrs, ["A" * subscript, "a" * subscript, "B" * subscript, "b" * subscript])
+        append!(
+            ltrs,
+            [
+                "A" * subscript,
+                "a" * subscript,
+                "B" * subscript,
+                "b" * subscript,
+            ],
+        )
    end
    Al = Alphabet(reverse!(ltrs))

@ -51,9 +59,13 @@ function SurfaceGroup(genus::Integer, boundaries::Integer, W=Word{Int16})
        comms = W(word)
        word_rels = [comms => one(comms)]

-        rws = let R = KnuthBendix.RewritingSystem(word_rels, KnuthBendix.Recursive(Al))
-            KnuthBendix.IndexAutomaton(KnuthBendix.knuthbendix(R))
-        end
+        rws =
+            let R = KnuthBendix.RewritingSystem(
+                    word_rels,
+                    KnuthBendix.Recursive(Al),
+                )
+                KnuthBendix.IndexAutomaton(KnuthBendix.knuthbendix(R))
+            end
    elseif boundaries == 1
        word_rels = Pair{W,W}[]
        rws = let R = RewritingSystem(word_rels, KnuthBendix.LenLex(Al))
@ -66,7 +78,13 @@ function SurfaceGroup(genus::Integer, boundaries::Integer, W=Word{Int16})
    F = FreeGroup(Al)
    rels = [F(lhs) => F(rhs) for (lhs, rhs) in word_rels]

-    return SurfaceGroup(genus, boundaries, [Al[i] for i in 2:2:length(Al)], rels, rws)
+    return SurfaceGroup(
+        genus,
+        boundaries,
+        [Al[i] for i in 2:2:length(Al)],
+        rels,
+        rws,
+    )
 end

 rewriting(S::SurfaceGroup) = S.rw
@ -75,17 +93,26 @@ relations(S::SurfaceGroup) = S.relations
 function symplectic_twists(π₁Σ::SurfaceGroup)
    g = genus(π₁Σ)

-    saut = SpecialAutomorphismGroup(FreeGroup(2g), max_rules=1000)
+    saut = SpecialAutomorphismGroup(FreeGroup(2g); max_rules = 1000)

-    Aij = [SymplecticMappingClass(saut, :A, i, j) for i in 1:g for j in 1:g if i ≠ j]
+    Aij = [
+        SymplecticMappingClass(saut, :A, i, j) for i in 1:g for
+        j in 1:g if i ≠ j
+    ]

-    Bij = [SymplecticMappingClass(saut, :B, i, j) for i in 1:g for j in 1:g if i ≠ j]
+    Bij = [
+        SymplecticMappingClass(saut, :B, i, j) for i in 1:g for
+        j in 1:g if i ≠ j
+    ]

-    mBij = [SymplecticMappingClass(saut, :B, i, j, minus=true) for i in 1:g for j in 1:g if i ≠ j]
+    mBij = [
+        SymplecticMappingClass(saut, :B, i, j; minus = true) for i in 1:g
+        for j in 1:g if i ≠ j
+    ]

    Bii = [SymplecticMappingClass(saut, :B, i, i) for i in 1:g]

-    mBii = [SymplecticMappingClass(saut, :B, i, i, minus=true) for i in 1:g]
+    mBii = [SymplecticMappingClass(saut, :B, i, i; minus = true) for i in 1:g]

    return [Aij; Bij; mBij; Bii; mBii]
 end
--- a/src/aut_groups/sautFn.jl
+++ b/src/aut_groups/sautFn.jl
@ -1,10 +1,13 @@
 include("transvections.jl")
 include("gersten_relations.jl")

-function SpecialAutomorphismGroup(F::FreeGroup; ordering=KnuthBendix.LenLex, kwargs...)
-
+function SpecialAutomorphismGroup(
+    F::FreeGroup;
+    ordering = KnuthBendix.LenLex,
+    kwargs...,
+)
    n = length(alphabet(F)) ÷ 2
-    A, rels = gersten_relations(n, commutative=false)
+    A, rels = gersten_relations(n; commutative = false)
    S = [A[i] for i in 1:2:length(A)]

    max_rules = 1000 * n
@ -12,7 +15,10 @@ function SpecialAutomorphismGroup(F::FreeGroup; ordering=KnuthBendix.LenLex, kwa
    rws = Logging.with_logger(Logging.NullLogger()) do
        rws = KnuthBendix.RewritingSystem(rels, ordering(A))
        # the rws is not confluent, let's suppress warning about it
-        KnuthBendix.knuthbendix(rws, KnuthBendix.Settings(; max_rules=max_rules, kwargs...))
+        return KnuthBendix.knuthbendix(
+            rws,
+            KnuthBendix.Settings(; max_rules = max_rules, kwargs...),
+        )
    end

    idxA = KnuthBendix.IndexAutomaton(rws)
@ -21,5 +27,5 @@ end

 function relations(G::AutomorphismGroup{<:FreeGroup})
    n = length(alphabet(object(G))) ÷ 2
-    return last(gersten_relations(n, commutative=false))
+    return last(gersten_relations(n; commutative = false))
 end
--- a/src/aut_groups/symplectic_twists.jl
+++ b/src/aut_groups/symplectic_twists.jl
@ -47,7 +47,15 @@ function Te_diagonal(λ::Groups.ΡΛ, ϱ::Groups.ΡΛ, i::Integer)
    return g
 end

-function Te_lantern(A::Alphabet, b₀::T, a₁::T, a₂::T, a₃::T, a₄::T, a₅::T) where {T}
+function Te_lantern(
+    A::Alphabet,
+    b₀::T,
+    a₁::T,
+    a₂::T,
+    a₃::T,
+    a₄::T,
+    a₅::T,
+) where {T}
    a₀ = (a₁ * a₂ * a₃)^4 * inv(b₀, A)
    X = a₄ * a₅ * a₃ * a₄ # from Primer
    b₁ = inv(X, A) * a₀ * X # from Primer
@ -85,7 +93,8 @@ function Te(λ::ΡΛ, ϱ::ΡΛ, i, j)
    if mod(j - (i + 1), genus) == 0
        return Te_diagonal(λ, ϱ, i)
    else
-        return inv(Te_lantern(
+        return inv(
+            Te_lantern(
                A,
                # Our notation:               # Primer notation:
                inv(Ta(λ, i + 1), A),         # b₀
@ -94,7 +103,9 @@ function Te(λ::ΡΛ, ϱ::ΡΛ, i, j)
                inv(Te_diagonal(λ, ϱ, i), A), # a₃
                inv(Tα(λ, i + 1), A),         # a₄
                inv(Te(λ, ϱ, i + 1, j), A),   # a₅
-            ), A)
+            ),
+            A,
+        )
    end
 end

@ -105,7 +116,6 @@ Return the element of `G` which corresponds to shifting generators of the free g
 In the corresponding mapping class group this element acts by rotation of the surface anti-clockwise.
 """
 function rotation_element(G::AutomorphismGroup{<:FreeGroup})
-
    A = alphabet(G)
    @assert iseven(ngens(object(G)))
    genus = ngens(object(G)) ÷ 2
@ -140,7 +150,10 @@ function rotation_element(λ::ΡΛ, ϱ::ΡΛ)
            Ta(λ, i) *
            inv(Te_diagonal(λ, ϱ, i), A)

-        Ta(λ, i) * inv(Ta(λ, j) * Tα(λ, j), A)^6 * (Ta(λ, j) * Tα(λ, j) * z)^4 * c
+        return Ta(λ, i) *
+               inv(Ta(λ, j) * Tα(λ, j), A)^6 *
+               (Ta(λ, j) * Tα(λ, j) * z)^4 *
+               c
    end

    τ = (Ta(λ, 1) * Tα(λ, 1))^6 * prod(halftwists)
@ -148,7 +161,6 @@ function rotation_element(λ::ΡΛ, ϱ::ΡΛ)
 end

 function mcg_twists(G::AutomorphismGroup{<:FreeGroup})
-
    @assert iseven(ngens(object(G)))
    genus = ngens(object(G)) ÷ 2

@ -178,7 +190,9 @@ struct SymplecticMappingClass{T,F} <: GSymbol
    f::F
 end

-Base.:(==)(a::SymplecticMappingClass, b::SymplecticMappingClass) = a.autFn_word == b.autFn_word
+function Base.:(==)(a::SymplecticMappingClass, b::SymplecticMappingClass)
+    return a.autFn_word == b.autFn_word
+end

 Base.hash(a::SymplecticMappingClass, h::UInt) = hash(a.autFn_word, h)

@ -187,8 +201,8 @@ function SymplecticMappingClass(
    id::Symbol,
    i::Integer,
    j::Integer;
-    minus=false,
-    inverse=false
+    minus = false,
+    inverse = false,
 )
    @assert i > 0 && j > 0
    id === :A && @assert i ≠ j
@ -246,7 +260,7 @@ function Base.show(io::IO, smc::SymplecticMappingClass)
    else
        print(io, smc.id, subscriptify(smc.i), ".", subscriptify(smc.j))
    end
-    smc.inv && print(io, "^-1")
+    return smc.inv && print(io, "^-1")
 end

 function Base.inv(m::SymplecticMappingClass)
@ -259,7 +273,7 @@ end
 function evaluate!(
    t::NTuple{N,T},
    smc::SymplecticMappingClass,
-    tmp=nothing,
+    tmp = nothing,
 ) where {N,T}
    t = smc.f(t)
    for i in 1:N
--- a/src/aut_groups/transvections.jl
+++ b/src/aut_groups/transvections.jl
@ -4,7 +4,7 @@ struct Transvection <: GSymbol
    j::UInt16
    inv::Bool

-    function Transvection(id::Symbol, i::Integer, j::Integer, inv=false)
+    function Transvection(id::Symbol, i::Integer, j::Integer, inv = false)
        @assert id in (:ϱ, :λ)
        return new(id, i, j, inv)
    end
@ -14,26 +14,24 @@ end
 λ(i, j) = Transvection(:λ, i, j)

 function Base.show(io::IO, t::Transvection)
-    id = if t.id === :ϱ
-        'ϱ'
-    else # if t.id === :λ
-        'λ'
-    end
-    print(io, id, subscriptify(t.i), '.', subscriptify(t.j))
-    t.inv && print(io, "^-1")
+    print(io, t.id, subscriptify(t.i), '.', subscriptify(t.j))
+    return t.inv && print(io, "^-1")
 end

 Base.inv(t::Transvection) = Transvection(t.id, t.i, t.j, !t.inv)

-Base.:(==)(t::Transvection, s::Transvection) =
-    t.id === s.id && t.i == s.i && t.j == s.j && t.inv == s.inv
+function Base.:(==)(t::Transvection, s::Transvection)
+    return t.id === s.id && t.i == s.i && t.j == s.j && t.inv == s.inv
+end

-Base.hash(t::Transvection, h::UInt) = hash(hash(t.id, hash(t.i)), hash(t.j, hash(t.inv, h)))
+function Base.hash(t::Transvection, h::UInt)
+    return hash(hash(t.id, hash(t.i)), hash(t.j, hash(t.inv, h)))
+end

 Base.@propagate_inbounds @inline function evaluate!(
    v::NTuple{T,N},
    t::Transvection,
-    tmp=one(first(v)),
+    tmp = one(first(v)),
 ) where {T,N}
    i, j = t.i, t.j
    @assert 1 ≤ i ≤ length(v) && 1 ≤ j ≤ length(v)
@ -84,7 +82,7 @@ end

 function Base.show(io::IO, p::PermRightAut)
    print(io, 'σ')
-    join(io, (subscriptify(Int(i)) for i in p.perm))
+    return join(io, (subscriptify(Int(i)) for i in p.perm))
 end

 Base.inv(p::PermRightAut) = PermRightAut(invperm(p.perm))
@ -92,4 +90,6 @@ Base.inv(p::PermRightAut) = PermRightAut(invperm(p.perm))
 Base.:(==)(p::PermRightAut, q::PermRightAut) = p.perm == q.perm
 Base.hash(p::PermRightAut, h::UInt) = hash(p.perm, hash(PermRightAut, h))

-evaluate!(v::NTuple{T,N}, p::PermRightAut, tmp=nothing) where {T,N} = v[p.perm]
+function evaluate!(v::NTuple{T,N}, p::PermRightAut, tmp = nothing) where {T,N}
+    return v[p.perm]
+end
--- a/src/autgroups.jl
+++ b/src/autgroups.jl
@ -1,10 +1,10 @@
 function KnuthBendix.Alphabet(S::AbstractVector{<:GSymbol})
-    S = unique!([S; inv.(S)])
+    S = union(S, inv.(S))
    inversions = [findfirst(==(inv(s)), S) for s in S]
    return Alphabet(S, inversions)
 end

-struct AutomorphismGroup{G<:Group,T,RW,S} <: AbstractFPGroup
+mutable struct AutomorphismGroup{G<:Group,T,RW,S} <: AbstractFPGroup
    group::G
    gens::Vector{T}
    rw::RW
@ -26,15 +26,18 @@ function equality_data(f::AbstractFPGroupElement{<:AutomorphismGroup})
    return imf
 end

-function Base.:(==)(g::A, h::A) where {A<:AbstractFPGroupElement{<:AutomorphismGroup}}
+function Base.:(==)(
+    g::A,
+    h::A,
+) where {A<:AbstractFPGroupElement{<:AutomorphismGroup}}
    @assert parent(g) === parent(h)

    if _isvalidhash(g) && _isvalidhash(h)
        hash(g) != hash(h) && return false
    end

-    length(word(g)) > 8 && normalform!(g)
-    length(word(h)) > 8 && normalform!(h)
+    normalform!(g)
+    normalform!(h)

    word(g) == word(h) && return true

@ -79,32 +82,46 @@ end

 # eye-candy

-Base.show(io::IO, ::Type{<:FPGroupElement{<:AutomorphismGroup{T}}}) where {T} =
-    print(io, "Automorphism{$T, …}")
+function Base.show(
+    io::IO,
+    ::Type{<:FPGroupElement{<:AutomorphismGroup{T}}},
+) where {T}
+    return print(io, "Automorphism{$T, …}")
+end

-Base.show(io::IO, A::AutomorphismGroup) = print(io, "automorphism group of ", object(A))
+function Base.show(io::IO, A::AutomorphismGroup)
+    return print(io, "automorphism group of ", object(A))
+end

-function Base.show(io::IO, ::MIME"text/plain", a::AbstractFPGroupElement{<:AutomorphismGroup})
+function Base.show(
+    io::IO,
+    ::MIME"text/plain",
+    a::AbstractFPGroupElement{<:AutomorphismGroup},
+)
    println(io, " ┌ $(a):")
    d = domain(a)
    im = evaluate(a)
    for (x, imx) in zip(d, im[1:end-1])
        println(io, " │   $x ↦ $imx")
    end
-    println(io, " └   $(last(d)) ↦ $(last(im))")
+    return println(io, " └   $(last(d)) ↦ $(last(im))")
 end

 ## Automorphism Evaluation

-domain(f::AbstractFPGroupElement{<:AutomorphismGroup}) = deepcopy(parent(f).domain)
+function domain(f::AbstractFPGroupElement{<:AutomorphismGroup})
+    return deepcopy(parent(f).domain)
+end
 # tuple(gens(object(parent(f)))...)

-evaluate(f::AbstractFPGroupElement{<:AutomorphismGroup}) = evaluate!(domain(f), f)
+function evaluate(f::AbstractFPGroupElement{<:AutomorphismGroup})
+    return evaluate!(domain(f), f)
+end

 function evaluate!(
    t::NTuple{N,T},
    f::AbstractFPGroupElement{<:AutomorphismGroup{<:Group}},
-    tmp=one(first(t)),
+    tmp = one(first(t)),
 ) where {N,T<:FPGroupElement}
    A = alphabet(f)
    for idx in word(f)
@ -113,47 +130,55 @@ function evaluate!(
    return t
 end

-evaluate!(t::NTuple{N,T}, s::GSymbol, tmp=nothing) where {N,T} = throw("you need to implement `evaluate!(::$(typeof(t)), ::$(typeof(s)), ::Alphabet, tmp=one(first(t)))`")
+function evaluate!(t::NTuple{N,T}, s::GSymbol, tmp = nothing) where {N,T}
+    throw(
+        "you need to implement `evaluate!(::$(typeof(t)), ::$(typeof(s)), ::Alphabet, tmp=one(first(t)))`",
+    )
+end

 # forward evaluate by substitution

-struct LettersMap{T,A}
-    indices_map::Dict{Int,T}
+struct LettersMap{W<:AbstractWord,A}
+    indices_map::Dict{Int,W}
    A::A
 end

 function LettersMap(a::FPGroupElement{<:AutomorphismGroup})
    dom = domain(a)
-    @assert all(isone ∘ length ∘ word, dom)
-    A = alphabet(first(dom))
-    first_letters = first.(word.(dom))
-    img = evaluate!(dom, a)
+    if all(isone ∘ length ∘ word, dom)
+        A = alphabet(first(dom))
+        first_letters = first.(word.(dom))
+        img = evaluate!(dom, a)

-    # (dom[i] → img[i] is a map from domain to images)
-    # we need a map from alphabet indices → (gens, gens⁻¹) → images
-    # here we do it for elements of the domain
-    # (trusting it's a set of generators that define a)
-    @assert length(dom) == length(img)
+        # (dom[i] → img[i] is a map from domain to images)
+        # we need a map from alphabet indices → (gens, gens⁻¹) → images
+        # here we do it for elements of the domain
+        # (trusting it's a set of generators that define a)
+        @assert length(dom) == length(img)

-    indices_map = Dict(A[A[fl]] => word(im) for (fl, im) in zip(first_letters, img))
-    # inverses of generators are dealt lazily in getindex
+        indices_map =
+            Dict(Int(fl) => word(im) for (fl, im) in zip(first_letters, img))
+        # inverses of generators are dealt lazily in getindex
+    else
+        throw("LettersMap is not implemented for non-generators in domain")
+    end

    return LettersMap(indices_map, A)
 end

-
-function Base.getindex(lm::LettersMap, i::Integer)
+function Base.getindex(lm::LettersMap{W}, i::Integer) where {W}
    # here i is an index of an alphabet
    @boundscheck 1 ≤ i ≤ length(lm.A)

    if !haskey(lm.indices_map, i)
-        img = if haskey(lm.indices_map, inv(i, lm.A))
-            inv(lm.indices_map[inv(i, lm.A)], lm.A)
+        I = inv(i, lm.A)
+        if haskey(lm.indices_map, I)
+            img = inv(lm.indices_map[I], lm.A)
+            lm.indices_map[i] = img
        else
-            @warn "LetterMap: neither $i nor its inverse has assigned value"
-            one(valtype(lm.indices_map))
+            lm.indices_map[i] = W([i])
+            lm.indices_map[I] = W([I])
        end
-        lm.indices_map[i] = img
    end
    return lm.indices_map[i]
 end
@ -164,9 +189,10 @@ function (a::FPGroupElement{<:AutomorphismGroup})(g::FPGroupElement)
    return parent(g)(img_w)
 end

-evaluate(w::AbstractWord, lm::LettersMap) = evaluate!(one(w), w, lm)
+evaluate(w::AbstractWord, lm::LettersMap) = evaluate!(similar(w), w, lm)

 function evaluate!(res::AbstractWord, w::AbstractWord, lm::LettersMap)
+    resize!(res, 0)
    for i in w
        append!(res, lm[i])
    end
--- a/src/constructions/direct_power.jl
+++ b/src/constructions/direct_power.jl
@ -12,25 +12,29 @@ struct DirectPowerElement{GEl,N,Gr<:GroupsCore.Group} <: GroupsCore.GroupElement
    parent::DirectPower{Gr,N,GEl}
 end

-DirectPowerElement(
+function DirectPowerElement(
    elts::AbstractVector{<:GroupsCore.GroupElement},
    G::DirectPower,
-) = DirectPowerElement(ntuple(i -> elts[i], _nfold(G)), G)
+)
+    return DirectPowerElement(ntuple(i -> elts[i], _nfold(G)), G)
+end

 _nfold(::DirectPower{Gr,N}) where {Gr,N} = N

-Base.one(G::DirectPower) =
-    DirectPowerElement(ntuple(_ -> one(G.group), _nfold(G)), G)
+function Base.one(G::DirectPower)
+    return DirectPowerElement(ntuple(_ -> one(G.group), _nfold(G)), G)
+end

-Base.eltype(::Type{<:DirectPower{Gr,N,GEl}}) where {Gr,N,GEl} =
-    DirectPowerElement{GEl,N,Gr}
+function Base.eltype(::Type{<:DirectPower{Gr,N,GEl}}) where {Gr,N,GEl}
+    return DirectPowerElement{GEl,N,Gr}
+end

 function Base.iterate(G::DirectPower)
    itr = Iterators.ProductIterator(ntuple(i -> G.group, _nfold(G)))
    res = iterate(itr)
    @assert res !== nothing
    elt = DirectPowerElement(first(res), G)
-    return elt, (iterator=itr, state=last(res))
+    return elt, (iterator = itr, state = last(res))
 end

 function Base.iterate(G::DirectPower, state)
@ -38,7 +42,7 @@ function Base.iterate(G::DirectPower, state)
    res = iterate(itr, st)
    res === nothing && return nothing
    elt = DirectPowerElement(first(res), G)
-    return elt, (iterator=itr, state=last(res))
+    return elt, (iterator = itr, state = last(res))
 end

 function Base.IteratorSize(::Type{<:DirectPower{Gr,N}}) where {Gr,N}
@ -49,20 +53,12 @@ end

 Base.size(G::DirectPower) = ntuple(_ -> length(G.group), _nfold(G))

-GroupsCore.order(::Type{I}, G::DirectPower) where {I<:Integer} =
-    convert(I, order(I, G.group)^_nfold(G))
+function GroupsCore.order(::Type{I}, G::DirectPower) where {I<:Integer}
+    return convert(I, order(I, G.group)^_nfold(G))
+end

 GroupsCore.ngens(G::DirectPower) = _nfold(G) * ngens(G.group)

-function GroupsCore.gens(G::DirectPower, i::Integer)
-    k = ngens(G.group)
-    ci = CartesianIndices((k, _nfold(G)))
-    @boundscheck checkbounds(ci, i)
-    r, c = Tuple(ci[i])
-    tup = ntuple(j -> j == c ? gens(G.group, r) : one(G.group), _nfold(G))
-    return DirectPowerElement(tup, G)
-end
-
 function GroupsCore.gens(G::DirectPower)
    N = _nfold(G)
    S = gens(G.group)
@ -73,24 +69,14 @@ end

 Base.isfinite(G::DirectPower) = isfinite(G.group)

-function Base.rand(
-    rng::Random.AbstractRNG,
-    rs::Random.SamplerTrivial{<:DirectPower},
-)
-    G = rs[]
-    return DirectPowerElement(rand(rng, G.group, _nfold(G)), G)
-end
-
 GroupsCore.parent(g::DirectPowerElement) = g.parent

-Base.:(==)(g::DirectPowerElement, h::DirectPowerElement) =
-    (parent(g) === parent(h) && g.elts == h.elts)
+function Base.:(==)(g::DirectPowerElement, h::DirectPowerElement)
+    return (parent(g) === parent(h) && g.elts == h.elts)
+end

 Base.hash(g::DirectPowerElement, h::UInt) = hash(g.elts, hash(parent(g), h))

-Base.deepcopy_internal(g::DirectPowerElement, stackdict::IdDict) =
-    DirectPowerElement(Base.deepcopy_internal(g.elts, stackdict), parent(g))
-
 Base.inv(g::DirectPowerElement) = DirectPowerElement(inv.(g.elts), parent(g))

 function Base.:(*)(g::DirectPowerElement, h::DirectPowerElement)
@ -98,15 +84,61 @@ function Base.:(*)(g::DirectPowerElement, h::DirectPowerElement)
    return DirectPowerElement(g.elts .* h.elts, parent(g))
 end

-GroupsCore.order(::Type{I}, g::DirectPowerElement) where {I<:Integer} =
-    convert(I, reduce(lcm, (order(I, h) for h in g.elts), init=one(I)))
+# to make sure that parents are never copied i.e.
+# g and deepcopy(g) share their parent
+Base.deepcopy_internal(G::DirectPower, ::IdDict) = G
+
+################## Implementing Group Interface Done!
+
+function GroupsCore.gens(G::DirectPower, i::Integer)
+    k = ngens(G.group)
+    ci = CartesianIndices((k, _nfold(G)))
+    @boundscheck checkbounds(ci, i)
+    r, c = Tuple(ci[i])
+    tup = ntuple(j -> j == c ? gens(G.group, r) : one(G.group), _nfold(G))
+    return DirectPowerElement(tup, G)
+end
+
+# Overloading rand: the PRA of GroupsCore is known for not performing
+# well on direct sums
+function Random.Sampler(
+    RNG::Type{<:Random.AbstractRNG},
+    G::DirectPower,
+    repetition::Random.Repetition = Val(Inf),
+)
+    return Random.SamplerTrivial(G)
+end
+
+function Base.rand(
+    rng::Random.AbstractRNG,
+    rs::Random.SamplerTrivial{<:DirectPower},
+)
+    G = rs[]
+    return DirectPowerElement(rand(rng, G.group, _nfold(G)), G)
+end
+
+function GroupsCore.order(::Type{I}, g::DirectPowerElement) where {I<:Integer}
+    return convert(I, reduce(lcm, (order(I, h) for h in g.elts); init = one(I)))
+end

 Base.isone(g::DirectPowerElement) = all(isone, g.elts)

 function Base.show(io::IO, G::DirectPower)
    n = _nfold(G)
    nn = n == 1 ? "1-st" : n == 2 ? "2-nd" : n == 3 ? "3-rd" : "$n-th"
-    print(io, "Direct $(nn) power of $(G.group)")
+    return print(io, "Direct $(nn) power of ", G.group)
+end
+
+function Base.show(io::IO, g::DirectPowerElement)
+    print(io, "( ")
+    join(io, g.elts, ", ")
+    return print(" )")
+end
+
+# convienience:
+Base.@propagate_inbounds function Base.getindex(
+    g::DirectPowerElement,
+    i::Integer,
+)
+    return g.elts[i]
 end
-Base.show(io::IO, g::DirectPowerElement) =
-    print(io, "( ", join(g.elts, ", "), " )")
--- a/src/constructions/direct_product.jl
+++ b/src/constructions/direct_product.jl
@ -14,18 +14,22 @@ end

 DirectProductElement(g, h, G::DirectProduct) = DirectProduct((g, h), G)

-Base.one(G::DirectProduct) =
-    DirectProductElement((one(G.first), one(G.last)), G)
+function Base.one(G::DirectProduct)
+    return DirectProductElement((one(G.first), one(G.last)), G)
+end

-Base.eltype(::Type{<:DirectProduct{Gt,Ht,GEl,HEl}}) where {Gt,Ht,GEl,HEl} =
-    DirectProductElement{GEl,HEl,Gt,Ht}
+function Base.eltype(
+    ::Type{<:DirectProduct{Gt,Ht,GEl,HEl}},
+) where {Gt,Ht,GEl,HEl}
+    return DirectProductElement{GEl,HEl,Gt,Ht}
+end

 function Base.iterate(G::DirectProduct)
    itr = Iterators.product(G.first, G.last)
    res = iterate(itr)
    @assert res !== nothing
    elt = DirectProductElement(first(res), G)
-    return elt, (iterator=itr, state=last(res))
+    return elt, (iterator = itr, state = last(res))
 end

 function Base.iterate(G::DirectProduct, state)
@ -33,7 +37,7 @@ function Base.iterate(G::DirectProduct, state)
    res = iterate(itr, st)
    res === nothing && return nothing
    elt = DirectProductElement(first(res), G)
-    return elt, (iterator=itr, state=last(res))
+    return elt, (iterator = itr, state = last(res))
 end

 function Base.IteratorSize(::Type{<:DirectProduct{Gt,Ht}}) where {Gt,Ht}
@ -50,21 +54,57 @@ end

 Base.size(G::DirectProduct) = (length(G.first), length(G.last))

-GroupsCore.order(::Type{I}, G::DirectProduct) where {I<:Integer} =
-    convert(I, order(I, G.first) * order(I, G.last))
+function GroupsCore.order(::Type{I}, G::DirectProduct) where {I<:Integer}
+    return convert(I, order(I, G.first) * order(I, G.last))
+end

 GroupsCore.ngens(G::DirectProduct) = ngens(G.first) + ngens(G.last)

 function GroupsCore.gens(G::DirectProduct)
-    gens_first = [DirectProductElement((g, one(G.last)), G) for g in gens(G.first)]
+    gens_first =
+        [DirectProductElement((g, one(G.last)), G) for g in gens(G.first)]

-    gens_last = [DirectProductElement((one(G.first), g), G) for g in gens(G.last)]
+    gens_last =
+        [DirectProductElement((one(G.first), g), G) for g in gens(G.last)]

    return [gens_first; gens_last]
 end

 Base.isfinite(G::DirectProduct) = isfinite(G.first) && isfinite(G.last)

+GroupsCore.parent(g::DirectProductElement) = g.parent
+
+function Base.:(==)(g::DirectProductElement, h::DirectProductElement)
+    return (parent(g) === parent(h) && g.elts == h.elts)
+end
+
+Base.hash(g::DirectProductElement, h::UInt) = hash(g.elts, hash(parent(g), h))
+
+function Base.inv(g::DirectProductElement)
+    return DirectProductElement(inv.(g.elts), parent(g))
+end
+
+function Base.:(*)(g::DirectProductElement, h::DirectProductElement)
+    @assert parent(g) === parent(h)
+    return DirectProductElement(g.elts .* h.elts, parent(g))
+end
+
+# to make sure that parents are never copied i.e.
+# g and deepcopy(g) share their parent
+Base.deepcopy_internal(G::DirectProduct, ::IdDict) = G
+
+################## Implementing Group Interface Done!
+
+# Overloading rand: the PRA of GroupsCore is known for not performing
+# well on direct sums
+function Random.Sampler(
+    RNG::Type{<:Random.AbstractRNG},
+    G::DirectProduct,
+    repetition::Random.Repetition = Val(Inf),
+)
+    return Random.SamplerTrivial(G)
+end
+
 function Base.rand(
    rng::Random.AbstractRNG,
    rs::Random.SamplerTrivial{<:DirectProduct},
@ -73,30 +113,26 @@ function Base.rand(
    return DirectProductElement((rand(rng, G.first), rand(rng, G.last)), G)
 end

-GroupsCore.parent(g::DirectProductElement) = g.parent
-
-Base.:(==)(g::DirectProductElement, h::DirectProductElement) =
-    (parent(g) === parent(h) && g.elts == h.elts)
-
-Base.hash(g::DirectProductElement, h::UInt) = hash(g.elts, hash(parent(g), h))
-
-Base.deepcopy_internal(g::DirectProductElement, stackdict::IdDict) =
-    DirectProductElement(Base.deepcopy_internal(g.elts, stackdict), parent(g))
-
-Base.inv(g::DirectProductElement) =
-    DirectProductElement(inv.(g.elts), parent(g))
-
-function Base.:(*)(g::DirectProductElement, h::DirectProductElement)
-    @assert parent(g) === parent(h)
-    return DirectProductElement(g.elts .* h.elts, parent(g))
+function GroupsCore.order(::Type{I}, g::DirectProductElement) where {I<:Integer}
+    return convert(I, lcm(order(I, first(g.elts)), order(I, last(g.elts))))
 end

-GroupsCore.order(::Type{I}, g::DirectProductElement) where {I<:Integer} =
-    convert(I, lcm(order(I, first(g.elts)), order(I, last(g.elts))))
-
 Base.isone(g::DirectProductElement) = all(isone, g.elts)

-Base.show(io::IO, G::DirectProduct) =
-    print(io, "Direct product of $(G.first) and $(G.last)")
-Base.show(io::IO, g::DirectProductElement) =
-    print(io, "( $(join(g.elts, ",")) )")
+function Base.show(io::IO, G::DirectProduct)
+    return print(io, "Direct product of ", G.first, " and ", G.last)
+end
+
+function Base.show(io::IO, g::DirectProductElement)
+    print(io, "( ")
+    join(io, g.elts, ", ")
+    return print(io, " )")
+end
+
+# convienience:
+Base.@propagate_inbounds function Base.getindex(
+    g::DirectProductElement,
+    i::Integer,
+)
+    return g.elts[i]
+end
--- a/src/constructions/wreath_product.jl
+++ b/src/constructions/wreath_product.jl
@ -1,4 +1,4 @@
-import PermutationGroups: AbstractPermutationGroup, AbstractPerm, degree, SymmetricGroup
+import PermutationGroups as PG

 """
    WreathProduct(G::Group, P::AbstractPermutationGroup) <: Group
@ -13,20 +13,20 @@ product is defined as
 where `m^σ` denotes the action (from the right) of the permutation `σ` on
 `d`-tuples of elements from `G`.
 """
-struct WreathProduct{DP<:DirectPower,PGr<:AbstractPermutationGroup} <:
+struct WreathProduct{DP<:DirectPower,PGr<:PG.AbstractPermutationGroup} <:
       GroupsCore.Group
    N::DP
    P::PGr

-    function WreathProduct(G::Group, P::AbstractPermutationGroup)
-        N = DirectPower{degree(P)}(G)
+    function WreathProduct(G::Group, P::PG.AbstractPermutationGroup)
+        N = DirectPower{PG.AP.degree(P)}(G)
        return new{typeof(N),typeof(P)}(N, P)
    end
 end

 struct WreathProductElement{
    DPEl<:DirectPowerElement,
-    PEl<:AbstractPerm,
+    PEl<:PG.AP.AbstractPermutation,
    Wr<:WreathProduct,
 } <: GroupsCore.GroupElement
    n::DPEl
@ -35,32 +35,36 @@ struct WreathProductElement{

    function WreathProductElement(
        n::DirectPowerElement,
-        p::AbstractPerm,
+        p::PG.AP.AbstractPermutation,
        W::WreathProduct,
    )
-        new{typeof(n),typeof(p),typeof(W)}(n, p, W)
+        return new{typeof(n),typeof(p),typeof(W)}(n, p, W)
    end
 end

 Base.one(W::WreathProduct) = WreathProductElement(one(W.N), one(W.P), W)

-Base.eltype(::Type{<:WreathProduct{DP,PGr}}) where {DP,PGr} =
-    WreathProductElement{eltype(DP),eltype(PGr),WreathProduct{DP,PGr}}
+function Base.eltype(::Type{<:WreathProduct{DP,PGr}}) where {DP,PGr}
+    return WreathProductElement{eltype(DP),eltype(PGr),WreathProduct{DP,PGr}}
+end

 function Base.iterate(G::WreathProduct)
    itr = Iterators.product(G.N, G.P)
    res = iterate(itr)
    @assert res !== nothing
-    elt = WreathProductElement(first(res)..., G)
-    return elt, (iterator=itr, state=last(res))
+    ab, st = res
+    (a, b) = ab
+    elt = WreathProductElement(a, b, G)
+    return elt, (itr, st)
 end

 function Base.iterate(G::WreathProduct, state)
-    itr, st = state.iterator, state.state
+    itr, st = state
    res = iterate(itr, st)
    res === nothing && return nothing
-    elt = WreathProductElement(first(res)..., G)
-    return elt, (iterator=itr, state=last(res))
+    (a::eltype(G.N), b::eltype(G.P)), st = res
+    elt = WreathProductElement(a, b, G)
+    return elt, (itr, st)
 end

 function Base.IteratorSize(::Type{<:WreathProduct{DP,PGr}}) where {DP,PGr}
@ -78,8 +82,9 @@ end

 Base.size(G::WreathProduct) = (length(G.N), length(G.P))

-GroupsCore.order(::Type{I}, G::WreathProduct) where {I<:Integer} =
-    convert(I, order(I, G.N) * order(I, G.P))
+function GroupsCore.order(::Type{I}, G::WreathProduct) where {I<:Integer}
+    return convert(I, order(I, G.N) * order(I, G.P))
+end

 function GroupsCore.gens(G::WreathProduct)
    N_gens = [WreathProductElement(n, one(G.P), G) for n in gens(G.N)]
@ -89,33 +94,22 @@ end

 Base.isfinite(G::WreathProduct) = isfinite(G.N) && isfinite(G.P)

-function Base.rand(
-    rng::Random.AbstractRNG,
-    rs::Random.SamplerTrivial{<:WreathProduct},
-)
-
-    G = rs[]
-    return WreathProductElement(rand(rng, G.N), rand(rng, G.P), G)
-end
-
 GroupsCore.parent(g::WreathProductElement) = g.parent

-Base.:(==)(g::WreathProductElement, h::WreathProductElement) =
-    parent(g) === parent(h) && g.n == h.n && g.p == h.p
-
-Base.hash(g::WreathProductElement, h::UInt) =
-    hash(g.n, hash(g.p, hash(g.parent, h)))
-
-function Base.deepcopy_internal(g::WreathProductElement, stackdict::IdDict)
-    return WreathProductElement(
-        Base.deepcopy_internal(g.n, stackdict),
-        Base.deepcopy_internal(g.p, stackdict),
-        parent(g),
-    )
+function Base.:(==)(g::WreathProductElement, h::WreathProductElement)
+    return parent(g) === parent(h) && g.n == h.n && g.p == h.p
 end

-_act(p::AbstractPerm, n::DirectPowerElement) =
-    DirectPowerElement(n.elts^p, parent(n))
+function Base.hash(g::WreathProductElement, h::UInt)
+    return hash(g.n, hash(g.p, hash(g.parent, h)))
+end
+
+function _act(p::PG.AP.AbstractPermutation, n::DirectPowerElement)
+    return DirectPowerElement(
+        ntuple(i -> n.elts[i^p], length(n.elts)),
+        parent(n),
+    )
+end

 function Base.inv(g::WreathProductElement)
    pinv = inv(g.p)
@ -127,10 +121,36 @@ function Base.:(*)(g::WreathProductElement, h::WreathProductElement)
    return WreathProductElement(g.n * _act(g.p, h.n), g.p * h.p, parent(g))
 end

+# to make sure that parents are never copied i.e.
+# g and deepcopy(g) share their parent
+Base.deepcopy_internal(G::WreathProduct, ::IdDict) = G
+
+################## Implementing Group Interface Done!
+
+# Overloading rand: the PRA of GroupsCore is known for not performing
+# well on direct sums
+function Random.Sampler(
+    RNG::Type{<:Random.AbstractRNG},
+    G::WreathProduct,
+    repetition::Random.Repetition = Val(Inf),
+)
+    return Random.SamplerTrivial(G)
+end
+
+function Base.rand(
+    rng::Random.AbstractRNG,
+    rs::Random.SamplerTrivial{<:WreathProduct},
+)
+    G = rs[]
+    return WreathProductElement(rand(rng, G.N), rand(rng, G.P), G)
+end
+
 Base.isone(g::WreathProductElement) = isone(g.n) && isone(g.p)

-Base.show(io::IO, G::WreathProduct) =
-    print(io, "Wreath product of $(G.N.group) by $(G.P)")
-Base.show(io::IO, g::WreathProductElement) = print(io, "( $(g.n)≀$(g.p) )")
+function Base.show(io::IO, G::WreathProduct)
+    return print(io, "Wreath product of ", G.N.group, " by ", G.P)
+end

-Base.copy(g::WreathProductElement) = WreathProductElement(g.n, g.p, parent(g))
+function Base.show(io::IO, g::WreathProductElement)
+    return print(io, "( ", g.n, "≀", g.p, " )")
+end
--- a/src/hashing.jl
+++ b/src/hashing.jl
@ -1,6 +1,6 @@
 ## Hashing

-equality_data(g::AbstractFPGroupElement) = (normalform!(g); word(g))
+equality_data(g::AbstractFPGroupElement) = word(g)

 bitget(h::UInt, n::Int) = Bool((h & (1 << n)) >> n)
 bitclear(h::UInt, n::Int) = h & ~(1 << n)
@ -20,8 +20,12 @@ _isvalidhash(g::AbstractFPGroupElement) = bitget(g.savedhash, 1)
 _setnormalform(h::UInt, v::Bool) = bitset(h, v, 0)
 _setvalidhash(h::UInt, v::Bool) = bitset(h, v, 1)

-_setnormalform!(g::AbstractFPGroupElement, v::Bool) = g.savedhash = _setnormalform(g.savedhash, v)
-_setvalidhash!(g::AbstractFPGroupElement, v::Bool) = g.savedhash = _setvalidhash(g.savedhash, v)
+function _setnormalform!(g::AbstractFPGroupElement, v::Bool)
+    return g.savedhash = _setnormalform(g.savedhash, v)
+end
+function _setvalidhash!(g::AbstractFPGroupElement, v::Bool)
+    return g.savedhash = _setvalidhash(g.savedhash, v)
+end

 # To update hash use this internal method, possibly only after computing the
 # normal form of `g`:
@ -33,6 +37,7 @@ function _update_savedhash!(g::AbstractFPGroupElement, data)
 end

 function Base.hash(g::AbstractFPGroupElement, h::UInt)
+    g = normalform!(g)
    _isvalidhash(g) || _update_savedhash!(g, equality_data(g))
    return hash(g.savedhash >> count_ones(__BITFLAGS_MASK), h)
 end
--- a/src/homomorphisms.jl
+++ b/src/homomorphisms.jl
@ -67,11 +67,13 @@ struct Homomorphism{Gr1,Gr2,I,W}
        f,
        source::AbstractFPGroup,
        target::AbstractFPGroup;
-        check=true
+        check = true,
    )
        A = alphabet(source)
-        dct = Dict(i => convert(word_type(target), f(i, source, target))
-                   for i in 1:length(A))
+        dct = Dict(
+            i => convert(word_type(target), f(i, source, target)) for
+            i in 1:length(A)
+        )
        I = eltype(word_type(source))
        W = word_type(target)
        hom = new{typeof(source),typeof(target),I,W}(dct, source, target)
@ -79,7 +81,6 @@ struct Homomorphism{Gr1,Gr2,I,W}
        if check
            @assert hom(one(source)) == one(target)
            for x in gens(source)
-
                @assert hom(x^-1) == hom(x)^-1

                for y in gens(source)
@ -111,4 +112,6 @@ function (h::Homomorphism)(g::AbstractFPGroupElement)
    return h.target(w)
 end

-Base.show(io::IO, h::Homomorphism) = print(io, "Homomorphism\n from : $(h.source)\n to   : $(h.target)")
+function Base.show(io::IO, h::Homomorphism)
+    return print(io, "Homomorphism\n from : $(h.source)\n to   : $(h.target)")
+end
--- a/src/iteration.jl
+++ b/src/iteration.jl
@ -1,3 +1,5 @@
+import OrderedCollections: OrderedSet
+
 mutable struct FPGroupIter{S,T,GEl}
    seen::S
    seen_iter_state::T
--- a/src/matrix_groups/MatrixGroups.jl
+++ b/src/matrix_groups/MatrixGroups.jl
@ -9,10 +9,11 @@ import GroupsCore.Random # GroupsCore rand
 using ..Groups
 using Groups.KnuthBendix

-export SpecialLinearGroup, SymplecticGroup
+export MatrixGroup, SpecialLinearGroup, SymplecticGroup

 include("abstract.jl")

+include("matrix_group.jl")
 include("SLn.jl")
 include("Spn.jl")

--- a/src/matrix_groups/SLn.jl
+++ b/src/matrix_groups/SLn.jl
@ -1,42 +1,35 @@
 include("eltary_matrices.jl")

-struct SpecialLinearGroup{N,T,R,A,S} <: MatrixGroup{N,T}
+struct SpecialLinearGroup{N,T,R,S} <: AbstractMatrixGroup{N,T}
    base_ring::R
-    alphabet::A
-    gens::S
+    alphabet::Alphabet{S}
+    gens::Vector{S}

    function SpecialLinearGroup{N}(base_ring) where {N}
-        S = [ElementaryMatrix{N}(i, j, one(base_ring)) for i in 1:N for j in 1:N if i ≠ j]
+        S = [
+            ElementaryMatrix{N}(i, j, one(base_ring)) for i in 1:N for
+            j in 1:N if i ≠ j
+        ]
        alphabet = Alphabet(S)

-        return new{
-            N,
-            eltype(base_ring),
-            typeof(base_ring),
-            typeof(alphabet),
-            typeof(S)
-        }(base_ring, alphabet, S)
+        T = eltype(base_ring)
+        R = typeof(base_ring)
+        St = eltype(S)
+
+        return new{N,T,R,St}(base_ring, alphabet, S)
    end
 end

-GroupsCore.ngens(SL::SpecialLinearGroup{N}) where {N} = N^2 - N
+GroupsCore.ngens(SL::SpecialLinearGroup) = length(SL.gens)

-Base.show(io::IO, SL::SpecialLinearGroup{N,T}) where {N,T} =
-    print(io, "special linear group of $N×$N matrices over $T")
+function Base.show(io::IO, ::SpecialLinearGroup{N,T}) where {N,T}
+    return print(io, "SL{$N,$T}")
+end

 function Base.show(
    io::IO,
    ::MIME"text/plain",
-    sl::Groups.AbstractFPGroupElement{<:SpecialLinearGroup{N}}
-) where {N}
-
-    Groups.normalform!(sl)
-
-    print(io, "SL{$N,$(eltype(sl))} matrix: ")
-    KnuthBendix.print_repr(io, word(sl), alphabet(sl))
-    println(io)
-    Base.print_array(io, matrix_repr(sl))
+    SL::SpecialLinearGroup{N,T},
+) where {N,T}
+    return print(io, "special linear group of $N×$N matrices over $T")
 end
-
-Base.show(io::IO, sl::Groups.AbstractFPGroupElement{<:SpecialLinearGroup}) =
-    KnuthBendix.print_repr(io, word(sl), alphabet(sl))
--- a/src/matrix_groups/Spn.jl
+++ b/src/matrix_groups/Spn.jl
@ -1,49 +1,44 @@
 include("eltary_symplectic.jl")

-struct SymplecticGroup{N,T,R,A,S} <: MatrixGroup{N,T}
+struct SymplecticGroup{N,T,R,S} <: AbstractMatrixGroup{N,T}
    base_ring::R
-    alphabet::A
-    gens::S
+    alphabet::Alphabet{S}
+    gens::Vector{S}

    function SymplecticGroup{N}(base_ring) where {N}
        S = symplectic_gens(N, eltype(base_ring))
        alphabet = Alphabet(S)

-        return new{
-            N,
-            eltype(base_ring),
-            typeof(base_ring),
-            typeof(alphabet),
-            typeof(S)
-        }(base_ring, alphabet, S)
+        T = eltype(base_ring)
+        R = typeof(base_ring)
+        St = eltype(S)
+
+        return new{N,T,R,St}(base_ring, alphabet, S)
    end
 end

 GroupsCore.ngens(Sp::SymplecticGroup) = length(Sp.gens)

-Base.show(io::IO, ::SymplecticGroup{N}) where {N} = print(io, "group of $N×$N symplectic matrices")
+Base.show(io::IO, ::SymplecticGroup{N,T}) where {N,T} = print(io, "Sp{$N,$T}")

-function Base.show(
-    io::IO,
-    ::MIME"text/plain",
-    sp::Groups.AbstractFPGroupElement{<:SymplecticGroup{N}}
-) where {N}
-    Groups.normalform!(sp)
-    print(io, "$N×$N symplectic matrix: ")
-    KnuthBendix.print_repr(io, word(sp), alphabet(sp))
-    println(io)
-    Base.print_array(io, matrix_repr(sp))
+function Base.show(io::IO, ::MIME"text/plain", ::SymplecticGroup{N}) where {N}
+    return print(io, "group of $N×$N symplectic matrices")
 end

-_offdiag_idcs(n) = ((i, j) for i in 1:n for j in 1:n if i ≠ j)
-
-function symplectic_gens(N, T=Int8)
+function symplectic_gens(N, T = Int8)
    iseven(N) || throw(ArgumentError("N needs to be even!"))
    n = N ÷ 2

-    a_ijs = [ElementarySymplectic{N}(:A, i, j, one(T)) for (i, j) in _offdiag_idcs(n)]
+    _offdiag_idcs(n) = ((i, j) for i in 1:n for j in 1:n if i ≠ j)
+
+    a_ijs = [
+        ElementarySymplectic{N}(:A, i, j, one(T)) for (i, j) in _offdiag_idcs(n)
+    ]
    b_is = [ElementarySymplectic{N}(:B, n + i, i, one(T)) for i in 1:n]
-    c_ijs = [ElementarySymplectic{N}(:B, n + i, j, one(T)) for (i, j) in _offdiag_idcs(n)]
+    c_ijs = [
+        ElementarySymplectic{N}(:B, n + i, j, one(T)) for
+        (i, j) in _offdiag_idcs(n)
+    ]

    S = [a_ijs; b_is; c_ijs]

@ -60,11 +55,16 @@ function _std_symplectic_form(m::AbstractMatrix)
    n = r ÷ 2
    𝕆 = zeros(eltype(m), n, n)
    𝕀 = one(eltype(m)) * LinearAlgebra.I
-    Ω = [𝕆 -𝕀
-        𝕀 𝕆]
+    Ω = [
+        𝕆 -𝕀
+        𝕀 𝕆
+    ]
    return Ω
 end

-function issymplectic(mat::M, Ω=_std_symplectic_form(mat)) where {M<:AbstractMatrix}
+function issymplectic(
+    mat::M,
+    Ω = _std_symplectic_form(mat),
+) where {M<:AbstractMatrix}
    return Ω == transpose(mat) * Ω * mat
 end
--- a/src/matrix_groups/abstract.jl
+++ b/src/matrix_groups/abstract.jl
@ -1,40 +1,89 @@
-abstract type MatrixGroup{N,T} <: Groups.AbstractFPGroup end
-const MatrixGroupElement{N,T} = Groups.AbstractFPGroupElement{<:MatrixGroup{N,T}}
+abstract type AbstractMatrixGroup{N,T} <: Groups.AbstractFPGroup end
+const MatrixGroupElement{N,T} =
+    Groups.AbstractFPGroupElement{<:AbstractMatrixGroup{N,T}}

-Base.isone(g::MatrixGroupElement{N,T}) where {N,T} =
-    isone(word(g)) || matrix_repr(g) == LinearAlgebra.I
-
-function Base.:(==)(m1::M1, m2::M2) where {M1<:MatrixGroupElement,M2<:MatrixGroupElement}
-    parent(m1) === parent(m2) || return false
-    word(m1) == word(m2) && return true
-    return matrix_repr(m1) == matrix_repr(m2)
+function Base.isone(g::MatrixGroupElement{N,T}) where {N,T}
+    return isone(word(g)) || isone(matrix(g))
 end

-Base.size(m::MatrixGroupElement{N}) where {N} = (N, N)
-Base.eltype(m::MatrixGroupElement{N,T}) where {N,T} = T
+function Base.:(==)(
+    m1::M1,
+    m2::M2,
+) where {M1<:MatrixGroupElement,M2<:MatrixGroupElement}
+    parent(m1) === parent(m2) || return false
+    word(m1) == word(m2) && return true
+    return matrix(m1) == matrix(m2)
+end
+
+Base.size(::MatrixGroupElement{N}) where {N} = (N, N)
+Base.size(::MatrixGroupElement{N}, d) where {N} = ifelse(d::Integer <= 2, N, 1)
+Base.eltype(::MatrixGroupElement{N,T}) where {N,T} = T

 # three structural assumptions about matrix groups
-Groups.word(sl::MatrixGroupElement) = sl.word
-Base.parent(sl::MatrixGroupElement) = sl.parent
-Groups.alphabet(M::MatrixGroup) = M.alphabet
-Groups.rewriting(M::MatrixGroup) = alphabet(M)
+Groups.word(m::MatrixGroupElement) = m.word
+Base.parent(m::MatrixGroupElement) = m.parent
+Groups.alphabet(M::AbstractMatrixGroup) = M.alphabet
+Groups.rewriting(M::AbstractMatrixGroup) = alphabet(M)

-Base.hash(sl::MatrixGroupElement, h::UInt) =
-    hash(matrix_repr(sl), hash(parent(sl), h))
+Base.hash(m::MatrixGroupElement, h::UInt) = hash(matrix(m), hash(parent(m), h))

-function matrix_repr(m::MatrixGroupElement{N,T}) where {N,T}
+function matrix(m::MatrixGroupElement{N,T}) where {N,T}
    if isone(word(m))
        return StaticArrays.SMatrix{N,N,T}(LinearAlgebra.I)
    end
    A = alphabet(parent(m))
-    return prod(matrix_repr(A[l]) for l in word(m))
+    return prod(matrix(A[l]) for l in word(m))
 end

+function Base.convert(
+    ::Type{M},
+    m::MatrixGroupElement,
+) where {M<:AbstractMatrix}
+    return convert(M, matrix(m))
+end
+(M::Type{<:AbstractMatrix})(m::MatrixGroupElement) = convert(M, m)
+
 function Base.rand(
    rng::Random.AbstractRNG,
-    rs::Random.SamplerTrivial{<:MatrixGroup},
+    rs::Random.SamplerTrivial{<:AbstractMatrixGroup},
 )
    Mgroup = rs[]
    S = gens(Mgroup)
-    return prod(g -> rand(Bool) ? g : inv(g), rand(S, rand(1:30)))
+    return prod(
+        g -> rand(rng, Bool) ? g : inv(g),
+        rand(rng, S, rand(rng, 1:30)),
+    )
+end
+
+function Base.show(io::IO, M::AbstractMatrixGroup)
+    g = gens(M, 1)
+    N = size(g, 1)
+    return print(io, "H ⩽ GL{$N,$(eltype(g))}")
+end
+
+function Base.show(io::IO, ::MIME"text/plain", M::AbstractMatrixGroup)
+    N = size(gens(M, 1), 1)
+    ng = GroupsCore.ngens(M)
+    return print(
+        io,
+        "subgroup of $N×$N invertible matrices with $(ng) generators",
+    )
+end
+
+function Base.show(
+    io::IO,
+    mat::Groups.AbstractFPGroupElement{<:AbstractMatrixGroup},
+)
+    return KnuthBendix.print_repr(io, word(mat), alphabet(mat))
+end
+
+function Base.show(
+    io::IO,
+    ::MIME"text/plain",
+    mat::Groups.AbstractFPGroupElement{<:AbstractMatrixGroup{N}},
+) where {N}
+    Groups.normalform!(mat)
+    KnuthBendix.print_repr(io, word(mat), alphabet(mat))
+    println(io, " ∈ ", parent(mat))
+    return Base.print_array(io, matrix(mat))
 end
--- a/src/matrix_groups/eltary_matrices.jl
+++ b/src/matrix_groups/eltary_matrices.jl
@ -2,25 +2,29 @@ struct ElementaryMatrix{N,T} <: Groups.GSymbol
    i::Int
    j::Int
    val::T
-    ElementaryMatrix{N}(i, j, val=1) where {N} =
-        (@assert i ≠ j; new{N,typeof(val)}(i, j, val))
+    function ElementaryMatrix{N}(i, j, val = 1) where {N}
+        return (@assert i ≠ j; new{N,typeof(val)}(i, j, val))
+    end
 end

 function Base.show(io::IO, e::ElementaryMatrix)
    print(io, 'E', Groups.subscriptify(e.i), Groups.subscriptify(e.j))
-    !isone(e.val) && print(io, "^$(e.val)")
+    return !isone(e.val) && print(io, "^$(e.val)")
 end

-Base.:(==)(e::ElementaryMatrix{N}, f::ElementaryMatrix{N}) where {N} =
-    e.i == f.i && e.j == f.j && e.val == f.val
+function Base.:(==)(e::ElementaryMatrix{N}, f::ElementaryMatrix{N}) where {N}
+    return e.i == f.i && e.j == f.j && e.val == f.val
+end

-Base.hash(e::ElementaryMatrix, h::UInt) =
-    hash(typeof(e), hash((e.i, e.j, e.val), h))
+function Base.hash(e::ElementaryMatrix, h::UInt)
+    return hash(typeof(e), hash((e.i, e.j, e.val), h))
+end

-Base.inv(e::ElementaryMatrix{N}) where {N} =
-    ElementaryMatrix{N}(e.i, e.j, -e.val)
+function Base.inv(e::ElementaryMatrix{N}) where {N}
+    return ElementaryMatrix{N}(e.i, e.j, -e.val)
+end

-function matrix_repr(e::ElementaryMatrix{N,T}) where {N,T}
+function matrix(e::ElementaryMatrix{N,T}) where {N,T}
    m = StaticArrays.MMatrix{N,N,T}(LinearAlgebra.I)
    m[e.i, e.j] = e.val
    x = StaticArrays.SMatrix{N,N}(m)
--- a/src/matrix_groups/eltary_symplectic.jl
+++ b/src/matrix_groups/eltary_symplectic.jl
@ -3,7 +3,12 @@ struct ElementarySymplectic{N,T} <: Groups.GSymbol
    i::Int
    j::Int
    val::T
-    function ElementarySymplectic{N}(s::Symbol, i::Integer, j::Integer, val=1) where {N}
+    function ElementarySymplectic{N}(
+        s::Symbol,
+        i::Integer,
+        j::Integer,
+        val = 1,
+    ) where {N}
        @assert s ∈ (:A, :B)
        @assert iseven(N)
        n = N ÷ 2
@ -18,8 +23,8 @@ end

 function Base.show(io::IO, s::ElementarySymplectic)
    i, j = Groups.subscriptify(s.i), Groups.subscriptify(s.j)
-    print(io, s.symbol, i, j)
-    !isone(s.val) && print(io, "^$(s.val)")
+    print(io, s.symbol, i, '.', j)
+    return !isone(s.val) && print(io, "^$(s.val)")
 end

 _ind(s::ElementarySymplectic{N}) where {N} = (s.i, s.j)
@ -41,23 +46,29 @@ function _dual_ind(N_half, i, j)
    return i, j
 end

-function Base.:(==)(s::ElementarySymplectic{N}, t::ElementarySymplectic{M}) where {N,M}
+function Base.:(==)(
+    s::ElementarySymplectic{N},
+    t::ElementarySymplectic{M},
+) where {N,M}
    N == M || return false
    s.symbol == t.symbol || return false
    s.val == t.val || return false
    return _ind(t) == _ind(s) || _ind(t) == _dual_ind(s)
 end

-Base.hash(s::ElementarySymplectic, h::UInt) =
-    hash(Set([_ind(s); _dual_ind(s)]), hash(s.symbol, hash(s.val, h)))
+function Base.hash(s::ElementarySymplectic, h::UInt)
+    return hash(Set([_ind(s); _dual_ind(s)]), hash(s.symbol, hash(s.val, h)))
+end

-LinearAlgebra.transpose(s::ElementarySymplectic{N}) where {N} =
-    ElementarySymplectic{N}(s.symbol, s.j, s.i, s.val)
+function LinearAlgebra.transpose(s::ElementarySymplectic{N}) where {N}
+    return ElementarySymplectic{N}(s.symbol, s.j, s.i, s.val)
+end

-Base.inv(s::ElementarySymplectic{N}) where {N} =
-    ElementarySymplectic{N}(s.symbol, s.i, s.j, -s.val)
+function Base.inv(s::ElementarySymplectic{N}) where {N}
+    return ElementarySymplectic{N}(s.symbol, s.i, s.j, -s.val)
+end

-function matrix_repr(s::ElementarySymplectic{N,T}) where {N,T}
+function matrix(s::ElementarySymplectic{N,T}) where {N,T}
    @assert iseven(N)
    n = div(N, 2)
    m = StaticArrays.MMatrix{N,N,T}(LinearAlgebra.I)
--- a/src/matrix_groups/matrix_generators.jl
+++ b/src/matrix_groups/matrix_generators.jl
@ -0,0 +1,36 @@
+struct MatrixElt{N,T,N²} <: Groups.GSymbol
+    id::Symbol
+    inv::Bool
+    mat::StaticArrays.SMatrix{N,N,T,N²}
+
+    function MatrixElt{N,T}(
+        id::Symbol,
+        mat::AbstractMatrix,
+        inv::Bool = false,
+    ) where {N,T}
+        n = LinearAlgebra.checksquare(mat)
+        @assert N == n
+        @assert !iszero(LinearAlgebra.det(mat))
+        return new{N,T,N^2}(id, inv, mat)
+    end
+end
+
+function MatrixElt{N}(
+    id::Symbol,
+    mat::AbstractMatrix,
+    inv::Bool = false,
+) where {N}
+    return MatrixElt{N,eltype(mat)}(id, mat, inv)
+end
+
+Base.show(io::IO, m::MatrixElt) = print(io, m.id, m.inv ? "⁻¹" : "")
+
+Base.:(==)(m::MatrixElt, n::MatrixElt) = m.mat == n.mat
+
+Base.hash(m::MatrixElt, h::UInt) = hash(m.mat, hash(typeof(m), h))
+
+function Base.inv(m::MatrixElt{N,T}) where {N,T}
+    return MatrixElt{N,T}(m.id, round.(T, inv(m.mat)), !m.inv)
+end
+
+matrix(m::MatrixElt) = m.mat
--- a/src/matrix_groups/matrix_group.jl
+++ b/src/matrix_groups/matrix_group.jl
@ -0,0 +1,25 @@
+include("matrix_generators.jl")
+
+struct MatrixGroup{N,T,R,S} <: AbstractMatrixGroup{N,T}
+    base_ring::R
+    alphabet::Alphabet{S}
+    gens::Vector{S}
+end
+
+function MatrixGroup{N}(
+    gens::AbstractVector{<:AbstractMatrix{T}},
+    base_ring = T,
+) where {N,T}
+    S = map(enumerate(gens)) do (i, mat)
+        id = Symbol('m', Groups.subscriptify(i))
+        return MatrixElt{N}(id, mat)
+    end
+    alphabet = Alphabet(S)
+
+    R = typeof(base_ring)
+    St = eltype(S)
+
+    return MatrixGroup{N,T,R,St}(base_ring, alphabet, S)
+end
+
+GroupsCore.ngens(M::MatrixGroup) = length(M.gens)
--- a/src/normalform.jl
+++ b/src/normalform.jl
@ -2,8 +2,10 @@
    normalform!(g::FPGroupElement)
 Compute the normal form of `g`, possibly modifying `g` in-place.
 """
-@inline function normalform!(g::AbstractFPGroupElement)
-    isnormalform(g) && return g
+@inline function normalform!(g::AbstractFPGroupElement; force = false)
+    if !force
+        isnormalform(g) && return g
+    end

    let w = one(word(g))
        w = normalform!(w, g)
@ -36,9 +38,9 @@ end

 """
    normalform!(res::AbstractWord, g::FPGroupElement)
-Append the normal form of `g` to word `res`, modifying `res` in place.
+Write the normal form of `g` to word `res`, modifying `res` in place.

-Defaults to the rewriting in the free group.
+The particular implementation of the normal form depends on `parent(g)`.
 """
@inline function normalform!(res::AbstractWord, g::AbstractFPGroupElement)
    isone(res) && isnormalform(g) && return append!(res, word(g))
--- a/src/rand.jl
+++ b/src/rand.jl
@ -0,0 +1,62 @@
+"""
+    PoissonSampler
+For a finitely presented group PoissonSampler returns group elements represented
+by words of length at most `R ~ Poisson(λ)` chosen uniformly at random.
+
+For finitely presented groups the Product Replacement Algorithm
+(see `PRASampler` from `GroupsCore.jl`) doesn't make much sense due to
+overly long words it produces. We therefore resort to a pseudo-random method,
+where a word `w` of length `R` is chosen uniformly at random among all
+words of length `R` where `R` follows the Poisson distribution.
+
+!!! note
+    Due to the choice of the parameters (`λ=8`) and the floating point
+    arithmetic the sampler will always return group elements represented by
+    words of length at most `42`.
+"""
+struct PoissonSampler{G,T} <: Random.Sampler{T}
+    group::G
+    λ::Int
+end
+
+function PoissonSampler(G::AbstractFPGroup; λ)
+    return PoissonSampler{typeof(G),eltype(G)}(G, λ)
+end
+
+function __poisson_invcdf(val; λ)
+    # __poisson_pdf(k, λ) = λ^k * ℯ^-λ / factorial(k)
+    # pdf = ntuple(k -> __poisson_pdf(k - 1, λ), 21)
+    # cdf = accumulate(+, pdf)
+    # radius = something(findfirst(>(val), cdf) - 1, 0)
+    # this is the iterative version:
+    pdf = ℯ^-λ
+    cdf = pdf
+    k = 0
+    while cdf < val
+        k += 1
+        pdf = pdf * λ / k
+        cdf += pdf
+    end
+    return k
+end
+
+function Random.rand(rng::Random.AbstractRNG, sampler::PoissonSampler)
+    R = __poisson_invcdf(rand(rng); λ = sampler.λ)
+
+    G = sampler.group
+    n = length(alphabet(G))
+    W = word_type(G)
+    T = eltype(W)
+
+    letters = rand(rng, T(1):T(n), R)
+    word = W(letters, false)
+    return G(word)
+end
+
+function Random.Sampler(
+    RNG::Type{<:Random.AbstractRNG},
+    G::AbstractFPGroup,
+    repetition::Random.Repetition = Val(Inf),
+)
+    return PoissonSampler(G; λ = 8)
+end
--- a/src/types.jl
+++ b/src/types.jl
@ -42,10 +42,7 @@ KnuthBendix.alphabet(G::AbstractFPGroup) = alphabet(ordering(G))
 Base.@propagate_inbounds function (G::AbstractFPGroup)(
    word::AbstractVector{<:Integer},
 )
-    @boundscheck @assert all(
-        l -> 1 <= l <= length(alphabet(G)),
-        word,
-    )
+    @boundscheck @assert all(l -> 1 <= l <= length(alphabet(G)), word)
    return FPGroupElement(word_type(G)(word), G)
 end

@ -53,8 +50,9 @@ end

 Base.one(G::AbstractFPGroup) = FPGroupElement(one(word_type(G)), G)

-Base.eltype(::Type{FPG}) where {FPG<:AbstractFPGroup} =
-    FPGroupElement{FPG,word_type(FPG)}
+function Base.eltype(::Type{FPG}) where {FPG<:AbstractFPGroup}
+    return FPGroupElement{FPG,word_type(FPG)}
+end

 include("iteration.jl")

@ -65,48 +63,45 @@ function GroupsCore.gens(G::AbstractFPGroup, i::Integer)
    l = alphabet(G)[G.gens[i]]
    return FPGroupElement(word_type(G)([l]), G)
 end
-GroupsCore.gens(G::AbstractFPGroup) =
-    [gens(G, i) for i in 1:GroupsCore.ngens(G)]
-
-# TODO: ProductReplacementAlgorithm
-function Base.rand(
-    rng::Random.AbstractRNG,
-    rs::Random.SamplerTrivial{<:AbstractFPGroup},
-)
-    l = rand(10:100)
-    G = rs[]
-    nletters = length(alphabet(G))
-    return FPGroupElement(word_type(G)(rand(1:nletters, l)), G)
+function GroupsCore.gens(G::AbstractFPGroup)
+    return [gens(G, i) for i in 1:GroupsCore.ngens(G)]
 end

-Base.isfinite(::AbstractFPGroup) = (
-    @warn "using generic isfinite(::AbstractFPGroup): the returned `false` might be wrong"; false
-)
+function Base.isfinite(::AbstractFPGroup)
+    return (
+        @warn "using generic isfinite(::AbstractFPGroup): the returned `false` might be wrong"; false
+    )
+end

 ## FPGroupElement

 abstract type AbstractFPGroupElement{Gr} <: GroupElement end

+Base.copy(g::AbstractFPGroupElement) = one(g) * g
+word(f::AbstractFPGroupElement) = f.word
+
 mutable struct FPGroupElement{Gr<:AbstractFPGroup,W<:AbstractWord} <:
               AbstractFPGroupElement{Gr}
    word::W
    savedhash::UInt
    parent::Gr

-    FPGroupElement(
+    function FPGroupElement(
        word::W,
        G::AbstractFPGroup,
-        hash::UInt=UInt(0),
-    ) where {W<:AbstractWord} = new{typeof(G),W}(word, hash, G)
+        hash::UInt = UInt(0),
+    ) where {W<:AbstractWord}
+        return new{typeof(G),W}(word, hash, G)
+    end

-    FPGroupElement{Gr,W}(word::AbstractWord, G::Gr) where {Gr,W} =
-        new{Gr,W}(word, UInt(0), G)
+    function FPGroupElement{Gr,W}(word::AbstractWord, G::Gr) where {Gr,W}
+        return new{Gr,W}(word, UInt(0), G)
+    end
 end

-Base.show(io::IO, ::Type{<:FPGroupElement{Gr}}) where {Gr} =
-    print(io, FPGroupElement, "{$Gr, …}")
-
-word(f::AbstractFPGroupElement) = f.word
+function Base.copy(f::FPGroupElement)
+    return FPGroupElement(copy(word(f)), parent(f), f.savedhash)
+end

 #convenience
 KnuthBendix.alphabet(g::AbstractFPGroupElement) = alphabet(parent(g))
@ -124,12 +119,33 @@ function Base.:(==)(g::AbstractFPGroupElement, h::AbstractFPGroupElement)
    @boundscheck @assert parent(g) === parent(h)
    normalform!(g)
    normalform!(h)
+    # I. compare hashes of the normalform
+    # II. compare some data associated to FPGroupElement,
+    # e.g. word, image of the domain etc.
    hash(g) != hash(h) && return false
-    return equality_data(g) == equality_data(h)
+    equality_data(g) == equality_data(h) && return true # compares
+
+    # if this failed it is still possible that the words together can be
+    # rewritten even further, so we
+    # 1. rewrite word(g⁻¹·h) w.r.t. rewriting(parent(g))
+    # 2. check if the result is empty
+    G = parent(g)
+
+    g⁻¹h = append!(inv(word(g), alphabet(G)), word(h))
+    # similar + empty preserve the storage size
+    # saves some re-allocations if res does not represent id
+    res = similar(word(g))
+    resize!(res, 0)
+    res = KnuthBendix.rewrite!(res, g⁻¹h, rewriting(G))
+    return isone(res)
 end

 function Base.deepcopy_internal(g::FPGroupElement, stackdict::IdDict)
-    return FPGroupElement(copy(word(g)), parent(g), g.savedhash)
+    haskey(stackdict, g) && return stackdict[g]
+    cw = Base.deepcopy_internal(word(g), stackdict)
+    h = FPGroupElement(cw, parent(g), g.savedhash)
+    stackdict[g] = h
+    return h
 end

 function Base.inv(g::GEl) where {GEl<:AbstractFPGroupElement}
@ -139,14 +155,26 @@ end

 function Base.:(*)(g::GEl, h::GEl) where {GEl<:AbstractFPGroupElement}
    @boundscheck @assert parent(g) === parent(h)
-    return GEl(word(g) * word(h), parent(g))
+    A = alphabet(parent(g))
+    k = 0
+    while k + 1 ≤ min(length(word(g)), length(word(h)))
+        if inv(word(g)[end-k], A) == word(h)[k+1]
+            k += 1
+        else
+            break
+        end
+    end
+    w = @view(word(g)[1:end-k]) * @view(word(h)[k+1:end])
+    res = GEl(w, parent(g))
+    return res
 end

-GroupsCore.isfiniteorder(g::AbstractFPGroupElement) =
-    isone(g) ? true :
-    (
+function GroupsCore.isfiniteorder(g::AbstractFPGroupElement)
+    return isone(g) ? true :
+           (
        @warn "using generic isfiniteorder(::AbstractFPGroupElement): the returned `false` might be wrong"; false
    )
+end

 # additional methods:
 Base.isone(g::AbstractFPGroupElement) = (normalform!(g); isempty(word(g)))
@ -164,12 +192,17 @@ struct FreeGroup{T,O} <: AbstractFPGroup
    end
 end

-FreeGroup(gens, A::Alphabet) = FreeGroup(gens, KnuthBendix.LenLex(A))
+function FreeGroup(n::Integer)
+    symbols =
+        collect(Iterators.flatten((Symbol(:f, i), Symbol(:F, i)) for i in 1:n))
+    inverses = collect(Iterators.flatten((2i, 2i - 1) for i in 1:n))
+    return FreeGroup(Alphabet(symbols, inverses))
+end

-function FreeGroup(A::Alphabet)
-    @boundscheck @assert all(
-        KnuthBendix.hasinverse(l, A) for l in A
-    )
+FreeGroup(A::Alphabet) = FreeGroup(KnuthBendix.LenLex(A))
+
+function __group_gens(A::Alphabet)
+    @boundscheck @assert all(KnuthBendix.hasinverse(l, A) for l in A)
    gens = Vector{eltype(A)}()
    invs = Vector{eltype(A)}()
    for l in A
@ -177,24 +210,17 @@ function FreeGroup(A::Alphabet)
        push!(gens, l)
        push!(invs, inv(l, A))
    end
-
-    return FreeGroup(gens, A)
+    return gens
 end

-function FreeGroup(n::Integer)
-    symbols = Symbol[]
-    inverses = Int[]
-    sizehint!(symbols, 2n)
-    sizehint!(inverses, 2n)
-    for i in 1:n
-        push!(symbols, Symbol(:f, i), Symbol(:F, i))
-        push!(inverses, 2i, 2i - 1)
-    end
-    return FreeGroup(symbols[1:2:2n], Alphabet(symbols, inverses))
+function FreeGroup(O::KnuthBendix.WordOrdering)
+    grp_gens = __group_gens(alphabet(O))
+    return FreeGroup(grp_gens, O)
 end

-Base.show(io::IO, F::FreeGroup) =
-    print(io, "free group on $(ngens(F)) generators")
+function Base.show(io::IO, F::FreeGroup)
+    return print(io, "free group on $(ngens(F)) generators")
+end

 # mandatory methods:
 KnuthBendix.ordering(F::FreeGroup) = F.ordering
@ -205,8 +231,9 @@ relations(F::FreeGroup) = Pair{eltype(F),eltype(F)}[]
 # these are mathematically correct
 Base.isfinite(::FreeGroup) = false

-GroupsCore.isfiniteorder(g::AbstractFPGroupElement{<:FreeGroup}) =
-    isone(g) ? true : false
+function GroupsCore.isfiniteorder(g::AbstractFPGroupElement{<:FreeGroup})
+    return isone(g) ? true : false
+end

 ## FP Groups

@ -222,8 +249,8 @@ rewriting(G::FPGroup) = G.rw
 function FPGroup(
    G::AbstractFPGroup,
    rels::AbstractVector{<:Pair{GEl,GEl}};
-    ordering=KnuthBendix.ordering(G),
-    kwargs...
+    ordering = KnuthBendix.ordering(G),
+    kwargs...,
 ) where {GEl<:FPGroupElement}
    for (lhs, rhs) in rels
        @assert parent(lhs) === parent(rhs) === G
@ -237,10 +264,18 @@ function FPGroup(
 end

 function Base.show(io::IO, ::MIME"text/plain", G::FPGroup)
-    print(io, "Finitely presented group generated by:\n\t{")
-    Base.print_array(io, permutedims(gens(G)))
-    println(io, " },")
-    println(io, "subject to relations:")
+    println(
+        io,
+        "Finitely presented group generated by $(ngens(G)) element",
+        ngens(G) > 1 ? 's' : "",
+        ": ",
+    )
+    join(io, gens(G), ", ", ", and ")
+    println(
+        io,
+        "\n    subject to relation",
+        length(relations(G)) > 1 ? 's' : "",
+    )
    return Base.print_array(io, relations(G))
 end

@ -253,8 +288,9 @@ function Base.show(io::IO, G::FPGroup)
    return print(io, " ⟩")
 end

-Base.show(io::IO, ::Type{<:FPGroup{T}}) where {T} =
-    print(io, FPGroup, "{$T, …}")
+function Base.show(io::IO, ::Type{<:FPGroup{T}}) where {T}
+    return print(io, FPGroup, "{$T, …}")
+end

 ## GSymbol aka letter of alphabet

--- a/src/wl_ball.jl
+++ b/src/wl_ball.jl
@ -1,6 +1,6 @@
 """
    wlmetric_ball(S::AbstractVector{<:GroupElem}
-        [, center=one(first(S)); radius=2, op=*, threading=true])
+        [, center=one(first(S)); radius=2, op=*])
 Compute metric ball as a list of elements of non-decreasing length, given the
 word-length metric on the group generated by `S`. The ball is centered at `center`
 (by default: the identity element). `radius` and `op` keywords specify the
@ -8,41 +8,29 @@ radius and multiplication operation to be used.
 """
 function wlmetric_ball(
    S::AbstractVector{T},
-    center::T=one(first(S));
-    radius=2,
-    op=*,
-    threading=true
+    center::T = one(first(S));
+    radius = 2,
+    op = *,
 ) where {T}
-    threading && return wlmetric_ball_thr(S, center, radius=radius, op=op)
-    return wlmetric_ball_serial(S, center, radius=radius, op=op)
-end
-
-function wlmetric_ball_serial(S::AbstractVector{T}, center::T=one(first(S)); radius=2, op=*) where {T}
-    @assert radius >= 1
-    old = union!([center], [center * s for s in S])
-    return _wlmetric_ball(S, old, radius, op, collect, unique!)
-end
-
-function wlmetric_ball_thr(S::AbstractVector{T}, center::T=one(first(S)); radius=2, op=*) where {T}
-    @assert radius >= 1
-    old = union!([center], [center * s for s in S])
-    return _wlmetric_ball(S, old, radius, op, Folds.collect, Folds.unique)
-end
-
-function _wlmetric_ball(S, old, radius, op, collect, unique)
-    sizes = [1, length(old)]
-    for r in 2:radius
-        old = let old = old, S = S,
-            new = collect(
-                (g = op(o, s); hash(g); g)
-                for o in @view(old[sizes[end-1]:end]) for s in S
-            )
-
-            append!(old, new)
-            unique(old)
+    ball = [center]
+    sizes = [1]
+    if radius ≤ 0
+        return ball, sizes[2:end]
+    else
+        ball = union!(ball, [center * s for s in S])
+        push!(sizes, length(ball))
+        if radius == 1
+            return ball, sizes[2:end]
+        else
+            for _ in 2:radius
+                new = collect(
+                    op(o, s) for o in @view(ball[sizes[end-1]:end]) for s in S
+                )
+                ball = union!(ball, new)
+                push!(sizes, length(ball))
+            end
        end
-        push!(sizes, length(old))
+        return ball, sizes[2:end]
    end
-    return old, sizes[2:end]
+    # return wlmetric_ball_serial(S, center; radius = radius, op = op)
 end
-
--- a/test/AutFn.jl
+++ b/test/AutFn.jl
@ -38,7 +38,7 @@
        [2, 1, 4, 3, 6, 5, 8, 7, 10, 9]
    )

-    F4 = FreeGroup([:a, :b, :c, :d], A4)
+    F4 = FreeGroup(A4)
    a, b, c, d = gens(F4)
    D = ntuple(i -> gens(F4, i), 4)

--- a/test/AutSigma3.jl
+++ b/test/AutSigma3.jl
@ -5,7 +5,7 @@

    @test contains(sprint(print, π₁Σ), "surface")

-    Groups.PermRightAut(p::Perm) = Groups.PermRightAut(p.d)
+    Groups.PermRightAut(p::Perm) = Groups.PermRightAut([i^p for i in 1:2genus])
    # Groups.PermLeftAut(p::Perm) = Groups.PermLeftAut(p.d)
    autπ₁Σ = let autπ₁Σ = AutomorphismGroup(π₁Σ)
        pauts = let p = perm"(1,3,5)(2,4,6)"
@ -50,8 +50,9 @@
    @test π₁Σ.(word.(z)) == Groups.domain(first(S))
    d = Groups.domain(first(S))
    p = perm"(1,3,5)(2,4,6)"
-    @test Groups.evaluate!(deepcopy(d), τ) == d^inv(p)
-    @test Groups.evaluate!(deepcopy(d), τ^2) == d^p
+    @test Groups.evaluate!(deepcopy(d), τ) ==
+          ntuple(i -> d[i^inv(p)], length(d))
+    @test Groups.evaluate!(deepcopy(d), τ^2) == ntuple(i -> d[i^p], length(d))

    E, sizes = Groups.wlmetric_ball(S, radius=3)
    @test sizes == [49, 1813, 62971]
--- a/test/benchmarks.jl
+++ b/test/benchmarks.jl
@ -5,14 +5,18 @@ using Groups

 function wl_ball(F; radius::Integer)
    g, state = iterate(F)
-    while length(word(g)) <= radius
+    sizes = Int[]
+    while length(sizes) ≤ radius
        res = iterate(F, state)
        isnothing(res) && break
        g, state = res
+        if length(word(g)) > length(sizes)
+            push!(sizes, length(state.seen) - 1)
+        end
    end
    elts = collect(state.seen)
-    elts = resize!(elts, length(elts)-1)
-    return elts
+    resize!(elts, sizes[end] - 1)
+    return elts, sizes[2:end]
 end

@testset "Benchmarks" begin
@ -25,21 +29,16 @@ end
        let G = FN
            S = unique([gens(G); inv.(gens(G))])

-            sizes1 = last(Groups.wlmetric_ball(S, radius=R, threading=false))
-            sizes2 = last(Groups.wlmetric_ball(S, radius=R, threading=true))
-
-            l = length(wl_ball(G, radius=R))
+            sizes1 = last(Groups.wlmetric_ball(S; radius = R))
+            sizes2 = last(wl_ball(G; radius = R))

            @test sizes1 == sizes2
-            @test last(sizes1) == l

-            @info "Ball of radius $R in $(parent(first(S)))" sizes=sizes1
+            @info "Ball of radius $R in $(parent(first(S)))" sizes = sizes1
            @info "serial"
-            @time Groups.wlmetric_ball(S, radius=R, threading=false)
-            @info "threaded"
-            @time Groups.wlmetric_ball(S, radius=R, threading=true)
+            @time Groups.wlmetric_ball(S, radius = R)
            @info "iteration"
-            @time wl_ball(G, radius=R)
+            @time wl_ball(G, radius = R)
        end
    end

@ -51,21 +50,16 @@ end
        let G = SAutFN
            S = unique([gens(G); inv.(gens(G))])

-            sizes1 = last(Groups.wlmetric_ball(S, radius=R, threading=false))
-            sizes2 = last(Groups.wlmetric_ball(S, radius=R, threading=true))
-
-            l = length(wl_ball(G, radius=R))
+            sizes1 = last(Groups.wlmetric_ball(S; radius = R))
+            sizes2 = last(wl_ball(G; radius = R))

            @test sizes1 == sizes2
-            @test last(sizes1) == l

-            @info "Ball of radius $R in $(parent(first(S)))" sizes=sizes1
+            @info "Ball of radius $R in $(parent(first(S)))" sizes = sizes1
            @info "serial"
-            @time Groups.wlmetric_ball(S, radius=R, threading=false)
-            @info "threaded"
-            @time Groups.wlmetric_ball(S, radius=R, threading=true)
+            @time Groups.wlmetric_ball(S, radius = R)
            @info "iteration"
-            @time wl_ball(G, radius=R)
+            @time wl_ball(G, radius = R)
        end
    end
 end
--- a/test/fp_groups.jl
+++ b/test/fp_groups.jl
@ -4,7 +4,7 @@
    @test FreeGroup(A) isa FreeGroup
    @test sprint(show, FreeGroup(A)) == "free group on 3 generators"

-    F = FreeGroup([:a, :b, :c], A)
+    F = FreeGroup([:a, :b, :c], Groups.KnuthBendix.LenLex(A))
    @test sprint(show, F) == "free group on 3 generators"

    a, b, c = gens(F)
@ -14,7 +14,8 @@
    G = FPGroup(F, [a * b => b * a, a * c => c * a, b * c => c * b])

    @test G isa FPGroup
-    @test sprint(show, G) == "⟨ a  b  c | \n\t  a*b => b*a  a*c => c*a  b*c => c*b ⟩"
+    @test sprint(show, G) ==
+          "⟨ a  b  c | \n\t  a*b => b*a  a*c => c*a  b*c => c*b ⟩"
    @test rand(G) isa FPGroupElement

    f = a * c * b
@ -40,7 +41,7 @@
    end

    # quotient of G
-    H = FPGroup(G, [aG^2 => cG, bG * cG => aG], max_rules=200)
+    H = FPGroup(G, [aG^2 => cG, bG * cG => aG]; max_rules = 200)

    h = H(word(g))

@ -48,15 +49,21 @@
    @test_throws AssertionError h == g
    @test_throws MethodError h * g

-    H′ = FPGroup(G, [aG^2 => cG, bG * cG => aG], max_rules=200)
+    H′ = FPGroup(G, [aG^2 => cG, bG * cG => aG]; max_rules = 200)
    @test_throws AssertionError one(H) == one(H′)

    Groups.normalform!(h)
    @test h == H([5])

-    @test_logs (:warn, "using generic isfiniteorder(::AbstractFPGroupElement): the returned `false` might be wrong") isfiniteorder(h)
+    @test_logs (
+        :warn,
+        "using generic isfiniteorder(::AbstractFPGroupElement): the returned `false` might be wrong",
+    ) isfiniteorder(h)

-    @test_logs (:warn, "using generic isfinite(::AbstractFPGroup): the returned `false` might be wrong") isfinite(H)
+    @test_logs (
+        :warn,
+        "using generic isfinite(::AbstractFPGroup): the returned `false` might be wrong",
+    ) isfinite(H)

    Logging.with_logger(Logging.NullLogger()) do
        @testset "GroupsCore conformance: H" begin
@ -64,4 +71,24 @@
            test_GroupElement_interface(rand(H, 2)...)
        end
    end
+
+    @testset "hash/normalform #28" begin
+        function cyclic_group(n::Integer)
+            A = Alphabet([:a, :A], [2, 1])
+            F = FreeGroup(A)
+            a, = Groups.gens(F)
+            e = one(F)
+            Cₙ = FPGroup(F, [a^n => e])
+
+            return Cₙ
+        end
+
+        n = 15
+        G = cyclic_group(n)
+        ball, sizes = Groups.wlmetric_ball(gens(G); radius = n)
+        @test first(sizes) == 2
+        @test last(sizes) == n
+
+        @test Set(ball) == Set([first(gens(G))^i for i in 0:n-1])
+    end
 end
--- a/test/free_groups.jl
+++ b/test/free_groups.jl
@ -1,7 +1,7 @@
@testset "FreeGroup" begin

    A3 = Alphabet([:a, :b, :c, :A, :B, :C], [4,5,6,1,2,3])
-    F3 = FreeGroup([:a, :b, :c], A3)
+    F3 = FreeGroup(A3)
    @test F3 isa FreeGroup

    @test gens(F3) isa Vector
--- a/test/group_constructions.jl
+++ b/test/group_constructions.jl
@ -1,9 +1,10 @@
@testset "GroupConstructions" begin

+    symmetric_group(n) = PermGroup(perm"(1,2)", Perm([2:n; 1]))
+
    @testset "DirectProduct" begin
        GH =
-            let G = PermutationGroups.SymmetricGroup(3),
-                H = PermutationGroups.SymmetricGroup(4)
+            let G = symmetric_group(3), H = symmetric_group(4)

                Groups.Constructions.DirectProduct(G, H)
            end
@ -17,7 +18,7 @@

    @testset "DirectPower" begin
        GGG = Groups.Constructions.DirectPower{3}(
-            PermutationGroups.SymmetricGroup(3),
+            symmetric_group(3),
        )
        test_Group_interface(GGG)
        test_GroupElement_interface(rand(GGG, 2)...)
@ -28,8 +29,7 @@
    end
    @testset "WreathProduct" begin
        W =
-            let G = PermutationGroups.SymmetricGroup(2),
-                P = PermutationGroups.SymmetricGroup(4)
+            let G = symmetric_group(2), P = symmetric_group(4)

                Groups.Constructions.WreathProduct(G, P)
            end
--- a/test/matrix_groups.jl
+++ b/test/matrix_groups.jl
@ -7,7 +7,7 @@ using Groups.MatrixGroups
        S = gens(SL3Z)
        union!(S, inv.(S))

-        _, sizes = Groups.wlmetric_ball(S, radius=4)
+        _, sizes = Groups.wlmetric_ball(S; radius = 4)

        @test sizes == [13, 121, 883, 5455]

@ -20,9 +20,7 @@ using Groups.MatrixGroups

        S = [w, r, s]
        S = unique([S; inv.(S)])
-        _, sizes = Groups.wlmetric_ball(S, radius=4)
-        @test sizes == [7, 33, 141, 561]
-        _, sizes = Groups.wlmetric_ball_serial(S, radius=4)
+        _, sizes = Groups.wlmetric_ball(S; radius = 4)
        @test sizes == [7, 33, 141, 561]

        Logging.with_logger(Logging.NullLogger()) do
@ -35,15 +33,18 @@ using Groups.MatrixGroups
            end
        end

+        x = w * inv(SL3Z(word(w)[end:end])) * r

-        x = w * inv(w) * r
-
-        @test length(word(x)) == 5
+        @test length(word(x)) == length(word(r))
        @test size(x) == (3, 3)
        @test eltype(x) == Int8

-        @test contains(sprint(print, SL3Z), "special linear group of 3×3")
-        @test contains(sprint(show, MIME"text/plain"(), x), "SL{3,Int8} matrix:")
+        @test contains(sprint(show, SL3Z), "SL{3,Int8}")
+        @test contains(
+            sprint(show, MIME"text/plain"(), SL3Z),
+            "special linear group",
+        )
+        @test contains(sprint(show, MIME"text/plain"(), x), "∈ SL{3,Int8}")
        @test sprint(print, x) isa String

        @test length(word(x)) == 3
@ -62,22 +63,57 @@ using Groups.MatrixGroups
            end
        end

-        @test contains(sprint(print, Sp6), "group of 6×6 symplectic matrices")
+        x = gens(Sp6, 1) * gens(Sp6, 2)^2
+        x *= inv(gens(Sp6, 2)^2) * gens(Sp6, 3)

-        x = gens(Sp6, 1)
-        x *= inv(x) * gens(Sp6, 2)
-
-        @test length(word(x)) == 3
+        @test length(word(x)) == 2
        @test size(x) == (6, 6)
        @test eltype(x) == Int8

-        @test contains(sprint(show, MIME"text/plain"(), x), "6×6 symplectic matrix:")
+        @test contains(sprint(show, Sp6), "Sp{6,Int8}")
+        @test contains(
+            sprint(show, MIME"text/plain"(), Sp6),
+            "group of 6×6 symplectic matrices",
+        )
+        @test contains(sprint(show, MIME"text/plain"(), x), "∈ Sp{6,Int8}")
+        @test sprint(print, x) isa String
+
+        @test length(word(x)) == 2
+
+        for g in gens(Sp6)
+            @test MatrixGroups.issymplectic(MatrixGroups.matrix(g))
+        end
+    end
+
+    @testset "General matrix group" begin
+        Sp6 = MatrixGroups.SymplecticGroup{6}(Int8)
+        G = Groups.MatrixGroup{6}(Matrix{Int16}.(gens(Sp6)))
+
+        Logging.with_logger(Logging.NullLogger()) do
+            @testset "GroupsCore conformance" begin
+                test_Group_interface(G)
+                g = G(rand(1:length(alphabet(G)), 10))
+                h = G(rand(1:length(alphabet(G)), 10))
+
+                test_GroupElement_interface(g, h)
+            end
+        end
+
+        x = gens(G, 1) * gens(G, 2)^3
+        x *= gens(G, 2)^-3
+
+        @test length(word(x)) == 1
+        @test size(x) == (6, 6)
+        @test eltype(x) == Int16
+
+        @test contains(sprint(show, G), "H ⩽ GL{6,Int16}")
+        @test contains(
+            sprint(show, MIME"text/plain"(), G),
+            "subgroup of 6×6 invertible matrices",
+        )
+        @test contains(sprint(show, MIME"text/plain"(), x), "∈ H ⩽ GL{6,Int16}")
        @test sprint(print, x) isa String

        @test length(word(x)) == 1
-
-        for g in gens(Sp6)
-            @test MatrixGroups.issymplectic(MatrixGroups.matrix_repr(g))
-        end
    end
 end
Author	SHA1	Message	Date
Marek Kaluba	1fbd7b875b	Merge pull request #31 from kalmarek/mk/update_to_PG_0.6 Update to PermutationGroups-0.6	2024-02-13 12:08:48 +01:00
Marek Kaluba	c13226a625	bump to v0.8	2024-02-13 11:16:04 +01:00
Marek Kaluba	7776ac4c6e	Merge branch 'master' into mk/update_to_PG_0.6	2024-02-12 14:04:07 +01:00
Marek Kaluba	6ca9497dab	tweak the printing of FPGroup	2024-02-12 12:38:28 +01:00
Marek Kaluba	f4d018f087	update constructions to PG-0.6	2024-02-12 12:37:33 +01:00
Marek Kaluba	add9a6f287	when everything fails in == try rewriting inv(g)*h	2024-02-12 12:36:04 +01:00
Marek Kaluba	ba6d58ec77	rewrite/fix deepcopy_internal	2024-02-12 12:33:57 +01:00
Marek Kaluba	a33b871754	simplify the constructors of the FreeGroup	2024-02-12 12:17:48 +01:00
Marek Kaluba	efbb4eada8	remove the show overload for Type{<:FPGroupElement}	2024-02-12 12:16:32 +01:00
Marek Kaluba	148a472dd2	add PoissonSampler for FPGroupElements Words of length following the Poisson(λ=8) distribution are chosen uniformly at random.	2024-02-12 12:16:03 +01:00
Marek Kaluba	5993cb328f	normalform! should be a no-op for normalized words	2024-02-12 11:36:17 +01:00
Marek Kaluba	8137e40998	make LettersMap a bit safer to use	2024-02-12 11:34:51 +01:00
Marek Kaluba	866e431c1a	Merge pull request #30 from kalmarek/mk/update_PG_0_4 update to PermutationGroups-0.4	2023-08-26 10:36:37 +02:00
Marek Kaluba	22cf6297a9	bump version to 0.7.8	2023-08-26 10:25:34 +02:00
Marek Kaluba	1a51a87771	update to PermutationGroups-0.4	2023-08-26 10:25:33 +02:00
Marek Kaluba	d385992e92	Merge pull request #29 from kalmarek/fix/#28_normalform_hashing Fix for #28: normalform & hashing	2023-06-12 17:16:55 +02:00
Marek Kaluba	c8805d6890	bump to 0.7.7	2023-05-29 17:41:47 +02:00
Marek Kaluba	3260e66d37	format test/fp_groups.jl	2023-05-29 17:40:56 +02:00
Marek Kaluba	eacb32af68	add separating dot between indices	2023-05-25 12:19:30 +02:00
Marek Kaluba	126c8bbc22	add test for hash/normalform	2023-05-25 11:58:32 +02:00
Marek Kaluba	93a841359b	add missing normalform! before hashing	2023-05-25 11:57:32 +02:00
Marek Kaluba	7230106bfc	Merge pull request #27 from kalmarek/enh/simplify_wlmetric simplify wlmetric_ball	2023-03-22 23:59:07 +01:00
Marek Kaluba	751850568c	make equality_data immutable operation	2023-03-22 21:45:04 +01:00
Marek Kaluba	038fc29b81	update benchmark on wl_ball	2023-03-22 21:44:33 +01:00
Marek Kaluba	c69eff1540	freely reduce words upon *	2023-03-22 21:44:09 +01:00
Marek Kaluba	a1bc334fb2	make AutomorphismGroup mutable parent field of an automorphism is now a pointer (i.e. 8 bytes)	2023-03-22 21:43:00 +01:00
Marek Kaluba	1f1e51917a	remove threaded wlmetric_ball * the threaded version was hardly faster * there was a memory leak (?) that was gone with -t 1 * simplifies the whole thing	2023-03-22 21:41:37 +01:00
Marek Kaluba	8bd3f7ede6	fix CI badge in readme	2023-03-15 23:00:50 +01:00
Marek Kaluba	448857ef03	Merge pull request #26 from kalmarek/enh/matrix_groups add support for general matrix groups	2023-03-15 19:25:07 +01:00
Marek Kaluba	af3913d085	more formatting	2023-03-15 19:07:14 +01:00
Marek Kaluba	3cdef72977	format aut_groups	2023-03-15 18:37:10 +01:00
Marek Kaluba	29af9659d9	format MatrixGroups module	2023-03-15 18:32:27 +01:00
Marek Kaluba	a0d2186477	add tests for MatrixGroups	2023-03-15 18:30:55 +01:00
Marek Kaluba	b52de81e06	update tests for show for SLn, Spn	2023-03-15 18:29:40 +01:00
Marek Kaluba	46c95dbb83	add general matrix groups	2023-03-15 18:26:45 +01:00
Marek Kaluba	40cd92e3c4	simplify parametrisation of SLn and Spn	2023-03-15 18:25:55 +01:00
Marek Kaluba	116439d074	move show functions to abstract.jl	2023-03-15 17:20:52 +01:00
Marek Kaluba	a380959614	add AbstractMatrixGroup	2023-03-15 17:11:22 +01:00
Marek Kaluba	f096a869b8	rename matrix_repr to simply matrix	2023-03-15 17:04:05 +01:00
Marek Kaluba	b7a6f4e952	bump version 0.7.5	2022-11-15 19:53:52 +01:00
Marek Kaluba	9006f8a4a1	make wl_ball thread safe: pre-compute normalforms!	2022-11-15 19:52:40 +01:00
Marek Kaluba	22c21706d8	switch to old CompatHelper the gh action JuliaRegisters/compathelper-action@1 is still not released	2022-10-17 14:57:10 +02:00