From 9006f8a4a1b583175a2fb420865a1f653beeaec0 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Tue, 15 Nov 2022 19:52:40 +0100
Subject: [PATCH 01/12] make wl_ball thread safe: pre-compute normalforms!

---
 src/wl_ball.jl | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/wl_ball.jl b/src/wl_ball.jl
index 18645cb..0c52080 100644
--- a/src/wl_ball.jl
+++ b/src/wl_ball.jl
@@ -34,7 +34,8 @@ function _wlmetric_ball(S, old, radius, op, collect, unique)
     for r in 2:radius
         old = let old = old, S = S,
             new = collect(
-                (g = op(o, s); hash(g); g)
+                (g = op(o, s); normalform!(g); hash(g); g)
+                # normalform! and hash are to make assure thread-safety of produced elts
                 for o in @view(old[sizes[end-1]:end]) for s in S
             )
 

From b7a6f4e9529c3e13dda3c86622e69932b72fd2a8 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Tue, 15 Nov 2022 19:53:52 +0100
Subject: [PATCH 02/12] bump version 0.7.5

---
 Project.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project.toml b/Project.toml
index d9251fa..6b177eb 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "Groups"
 uuid = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
 authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
-version = "0.7.4"
+version = "0.7.5"
 
 [deps]
 Folds = "41a02a25-b8f0-4f67-bc48-60067656b558"

From f096a869b8cf4438e07437608a54c5029bdde298 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 16:51:22 +0100
Subject: [PATCH 03/12] rename matrix_repr to simply matrix

---
 src/abelianize.jl                      |  4 ++--
 src/matrix_groups/abstract.jl          | 12 ++++++------
 src/matrix_groups/eltary_matrices.jl   |  2 +-
 src/matrix_groups/eltary_symplectic.jl |  2 +-
 4 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/src/abelianize.jl b/src/abelianize.jl
index 4f47f0f..1ab92ee 100644
--- a/src/abelianize.jl
+++ b/src/abelianize.jl
@@ -53,7 +53,7 @@ function Groups._abelianize(
     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 +63,7 @@ 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
diff --git a/src/matrix_groups/abstract.jl b/src/matrix_groups/abstract.jl
index 1da2228..5e022f7 100644
--- a/src/matrix_groups/abstract.jl
+++ b/src/matrix_groups/abstract.jl
@@ -2,12 +2,12 @@ abstract type MatrixGroup{N,T} <: Groups.AbstractFPGroup end
 const MatrixGroupElement{N,T} = Groups.AbstractFPGroupElement{<:MatrixGroup{N,T}}
 
 Base.isone(g::MatrixGroupElement{N,T}) where {N,T} =
-    isone(word(g)) || matrix_repr(g) == LinearAlgebra.I
+    isone(word(g)) || isone(matrix(g))
 
 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)
+    return matrix(m1) == matrix(m2)
 end
 
 Base.size(m::MatrixGroupElement{N}) where {N} = (N, N)
@@ -19,15 +19,15 @@ Base.parent(sl::MatrixGroupElement) = sl.parent
 Groups.alphabet(M::MatrixGroup) = M.alphabet
 Groups.rewriting(M::MatrixGroup) = 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.rand(
diff --git a/src/matrix_groups/eltary_matrices.jl b/src/matrix_groups/eltary_matrices.jl
index 6e3ed48..42b582d 100644
--- a/src/matrix_groups/eltary_matrices.jl
+++ b/src/matrix_groups/eltary_matrices.jl
@@ -20,7 +20,7 @@ Base.hash(e::ElementaryMatrix, h::UInt) =
 Base.inv(e::ElementaryMatrix{N}) where {N} =
     ElementaryMatrix{N}(e.i, e.j, -e.val)
 
-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)
diff --git a/src/matrix_groups/eltary_symplectic.jl b/src/matrix_groups/eltary_symplectic.jl
index 0640e6b..91ab941 100644
--- a/src/matrix_groups/eltary_symplectic.jl
+++ b/src/matrix_groups/eltary_symplectic.jl
@@ -57,7 +57,7 @@ LinearAlgebra.transpose(s::ElementarySymplectic{N}) where {N} =
 Base.inv(s::ElementarySymplectic{N}) where {N} =
     ElementarySymplectic{N}(s.symbol, s.i, s.j, -s.val)
 
-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)

From a3809596144af510b47703e275a7b10eb5631f0b Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 17:11:22 +0100
Subject: [PATCH 04/12] add AbstractMatrixGroup

---
 src/matrix_groups/SLn.jl      | 2 +-
 src/matrix_groups/Spn.jl      | 2 +-
 src/matrix_groups/abstract.jl | 8 ++++----
 3 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/src/matrix_groups/SLn.jl b/src/matrix_groups/SLn.jl
index 5a783fa..72b9504 100644
--- a/src/matrix_groups/SLn.jl
+++ b/src/matrix_groups/SLn.jl
@@ -1,6 +1,6 @@
 include("eltary_matrices.jl")
 
-struct SpecialLinearGroup{N,T,R,A,S} <: MatrixGroup{N,T}
+struct SpecialLinearGroup{N,T,R,A,S} <: AbstractMatrixGroup{N,T}
     base_ring::R
     alphabet::A
     gens::S
diff --git a/src/matrix_groups/Spn.jl b/src/matrix_groups/Spn.jl
index 6d98bae..98f556a 100644
--- a/src/matrix_groups/Spn.jl
+++ b/src/matrix_groups/Spn.jl
@@ -1,6 +1,6 @@
 include("eltary_symplectic.jl")
 
-struct SymplecticGroup{N,T,R,A,S} <: MatrixGroup{N,T}
+struct SymplecticGroup{N,T,R,A,S} <: AbstractMatrixGroup{N,T}
     base_ring::R
     alphabet::A
     gens::S
diff --git a/src/matrix_groups/abstract.jl b/src/matrix_groups/abstract.jl
index 5e022f7..d90613b 100644
--- a/src/matrix_groups/abstract.jl
+++ b/src/matrix_groups/abstract.jl
@@ -1,5 +1,5 @@
-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)) || isone(matrix(g))
@@ -32,9 +32,9 @@ end
 
 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

From 116439d074914ae5ddef5341fe0cf80ca12564d7 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 17:20:52 +0100
Subject: [PATCH 05/12] move show functions to abstract.jl

---
 src/matrix_groups/SLn.jl      | 20 ++++++--------------
 src/matrix_groups/Spn.jl      | 15 +++------------
 src/matrix_groups/abstract.jl | 26 ++++++++++++++++++++++++++
 3 files changed, 35 insertions(+), 26 deletions(-)

diff --git a/src/matrix_groups/SLn.jl b/src/matrix_groups/SLn.jl
index 72b9504..d95107e 100644
--- a/src/matrix_groups/SLn.jl
+++ b/src/matrix_groups/SLn.jl
@@ -21,22 +21,14 @@ end
 
 GroupsCore.ngens(SL::SpecialLinearGroup{N}) where {N} = N^2 - N
 
-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))
diff --git a/src/matrix_groups/Spn.jl b/src/matrix_groups/Spn.jl
index 98f556a..d66c0af 100644
--- a/src/matrix_groups/Spn.jl
+++ b/src/matrix_groups/Spn.jl
@@ -21,20 +21,11 @@ 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)
diff --git a/src/matrix_groups/abstract.jl b/src/matrix_groups/abstract.jl
index d90613b..3caca2c 100644
--- a/src/matrix_groups/abstract.jl
+++ b/src/matrix_groups/abstract.jl
@@ -38,3 +38,29 @@ function Base.rand(
     S = gens(Mgroup)
     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)
+    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)
+    print(io, "subgroup of $N×$N invertible matrices with $(ng) generators")
+end
+
+Base.show(io::IO, mat::Groups.AbstractFPGroupElement{<:AbstractMatrixGroup}) =
+    KnuthBendix.print_repr(io, word(mat), alphabet(mat))
+
+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))
+    Base.print_array(io, matrix(mat))
+end

From 40cd92e3c427702c6d01103e20768db3ed288767 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 18:25:55 +0100
Subject: [PATCH 06/12] simplify parametrisation of SLn and Spn

---
 src/matrix_groups/SLn.jl | 25 +++++++++++++------------
 src/matrix_groups/Spn.jl | 18 ++++++++----------
 2 files changed, 21 insertions(+), 22 deletions(-)

diff --git a/src/matrix_groups/SLn.jl b/src/matrix_groups/SLn.jl
index d95107e..57e4f22 100644
--- a/src/matrix_groups/SLn.jl
+++ b/src/matrix_groups/SLn.jl
@@ -1,25 +1,26 @@
 include("eltary_matrices.jl")
 
-struct SpecialLinearGroup{N,T,R,A,S} <: AbstractMatrixGroup{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)
 
 function Base.show(io::IO, ::SpecialLinearGroup{N,T}) where {N,T}
     return print(io, "SL{$N,$T}")
diff --git a/src/matrix_groups/Spn.jl b/src/matrix_groups/Spn.jl
index d66c0af..2361050 100644
--- a/src/matrix_groups/Spn.jl
+++ b/src/matrix_groups/Spn.jl
@@ -1,21 +1,19 @@
 include("eltary_symplectic.jl")
 
-struct SymplecticGroup{N,T,R,A,S} <: AbstractMatrixGroup{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
 

From 46c95dbb83d409029ef39d9e9edc483c9473e489 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 18:26:45 +0100
Subject: [PATCH 07/12] add general matrix groups

---
 src/matrix_groups/MatrixGroups.jl      |  3 ++-
 src/matrix_groups/matrix_generators.jl | 36 ++++++++++++++++++++++++++
 src/matrix_groups/matrix_group.jl      | 25 ++++++++++++++++++
 3 files changed, 63 insertions(+), 1 deletion(-)
 create mode 100644 src/matrix_groups/matrix_generators.jl
 create mode 100644 src/matrix_groups/matrix_group.jl

diff --git a/src/matrix_groups/MatrixGroups.jl b/src/matrix_groups/MatrixGroups.jl
index 486baf1..64ad756 100644
--- 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")
 
diff --git a/src/matrix_groups/matrix_generators.jl b/src/matrix_groups/matrix_generators.jl
new file mode 100644
index 0000000..2a4975c
--- /dev/null
+++ 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
diff --git a/src/matrix_groups/matrix_group.jl b/src/matrix_groups/matrix_group.jl
new file mode 100644
index 0000000..51355db
--- /dev/null
+++ 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)

From b52de81e060cc3ff1c5d60ed385b6a1c68a162f6 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 18:28:10 +0100
Subject: [PATCH 08/12] update tests for show for SLn, Spn

---
 test/matrix_groups.jl | 26 ++++++++++++++++----------
 1 file changed, 16 insertions(+), 10 deletions(-)

diff --git a/test/matrix_groups.jl b/test/matrix_groups.jl
index ef27c9d..4e99f04 100644
--- 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 +20,9 @@ using Groups.MatrixGroups
 
         S = [w, r, s]
         S = unique([S; inv.(S)])
-        _, sizes = Groups.wlmetric_ball(S, radius=4)
+        _, 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_serial(S; radius = 4)
         @test sizes == [7, 33, 141, 561]
 
         Logging.with_logger(Logging.NullLogger()) do
@@ -35,15 +35,18 @@ using Groups.MatrixGroups
             end
         end
 
-
         x = w * inv(w) * r
 
         @test length(word(x)) == 5
         @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
@@ -62,8 +65,6 @@ using Groups.MatrixGroups
             end
         end
 
-        @test contains(sprint(print, Sp6), "group of 6×6 symplectic matrices")
-
         x = gens(Sp6, 1)
         x *= inv(x) * gens(Sp6, 2)
 
@@ -71,13 +72,18 @@ using Groups.MatrixGroups
         @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)) == 1
 
         for g in gens(Sp6)
-            @test MatrixGroups.issymplectic(MatrixGroups.matrix_repr(g))
+            @test MatrixGroups.issymplectic(MatrixGroups.matrix(g))
         end
     end
 end

From a0d2186477dae8aaad5523f1601dba8389ce532b Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 18:30:55 +0100
Subject: [PATCH 09/12] add tests for MatrixGroups

---
 test/matrix_groups.jl | 32 ++++++++++++++++++++++++++++++++
 1 file changed, 32 insertions(+)

diff --git a/test/matrix_groups.jl b/test/matrix_groups.jl
index 4e99f04..e93c70e 100644
--- a/test/matrix_groups.jl
+++ b/test/matrix_groups.jl
@@ -86,4 +86,36 @@ using Groups.MatrixGroups
             @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)
+        x *= inv(x) * gens(G, 2)
+
+        @test length(word(x)) == 3
+        @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
+    end
 end

From 29af9659d96c484622094500eb9db294de8fe6e7 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 18:32:27 +0100
Subject: [PATCH 10/12] format MatrixGroups module

---
 src/matrix_groups/Spn.jl               | 25 ++++++++---
 src/matrix_groups/abstract.jl          | 61 ++++++++++++++++++--------
 src/matrix_groups/eltary_matrices.jl   | 22 ++++++----
 src/matrix_groups/eltary_symplectic.jl | 29 ++++++++----
 4 files changed, 93 insertions(+), 44 deletions(-)

diff --git a/src/matrix_groups/Spn.jl b/src/matrix_groups/Spn.jl
index 2361050..7bcfbce 100644
--- a/src/matrix_groups/Spn.jl
+++ b/src/matrix_groups/Spn.jl
@@ -24,15 +24,21 @@ Base.show(io::IO, ::SymplecticGroup{N,T}) where {N,T} = print(io, "Sp{$N,$T}")
 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]
 
@@ -49,11 +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
diff --git a/src/matrix_groups/abstract.jl b/src/matrix_groups/abstract.jl
index 3caca2c..ac2b625 100644
--- a/src/matrix_groups/abstract.jl
+++ b/src/matrix_groups/abstract.jl
@@ -1,26 +1,31 @@
 abstract type AbstractMatrixGroup{N,T} <: Groups.AbstractFPGroup end
-const MatrixGroupElement{N,T} = Groups.AbstractFPGroupElement{<:AbstractMatrixGroup{N,T}}
+const MatrixGroupElement{N,T} =
+    Groups.AbstractFPGroupElement{<:AbstractMatrixGroup{N,T}}
 
-Base.isone(g::MatrixGroupElement{N,T}) where {N,T} =
-    isone(word(g)) || isone(matrix(g))
+function Base.isone(g::MatrixGroupElement{N,T}) where {N,T}
+    return isone(word(g)) || isone(matrix(g))
+end
 
-function Base.:(==)(m1::M1, m2::M2) where {M1<:MatrixGroupElement,M2<:MatrixGroupElement}
+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(m::MatrixGroupElement{N}) where {N} = (N, N)
-Base.eltype(m::MatrixGroupElement{N,T}) where {N,T} = T
+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(m::MatrixGroupElement, h::UInt) =
-    hash(matrix(m), hash(parent(m), h))
+Base.hash(m::MatrixGroupElement, h::UInt) = hash(matrix(m), hash(parent(m), h))
 
 function matrix(m::MatrixGroupElement{N,T}) where {N,T}
     if isone(word(m))
@@ -30,37 +35,55 @@ function matrix(m::MatrixGroupElement{N,T}) where {N,T}
     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{<:AbstractMatrixGroup},
 )
     Mgroup = rs[]
     S = gens(Mgroup)
-    return prod(g -> rand(rng, Bool) ? g : inv(g), rand(rng, S, rand(rng, 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)
-    print(io, "H ⩽ GL{$N,$(eltype(g))}")
+    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)
-    print(io, "subgroup of $N×$N invertible matrices with $(ng) generators")
+    return print(
+        io,
+        "subgroup of $N×$N invertible matrices with $(ng) generators",
+    )
 end
 
-Base.show(io::IO, mat::Groups.AbstractFPGroupElement{<:AbstractMatrixGroup}) =
-    KnuthBendix.print_repr(io, word(mat), alphabet(mat))
+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}}
+    mat::Groups.AbstractFPGroupElement{<:AbstractMatrixGroup{N}},
 ) where {N}
     Groups.normalform!(mat)
     KnuthBendix.print_repr(io, word(mat), alphabet(mat))
     println(io, " ∈ ", parent(mat))
-    Base.print_array(io, matrix(mat))
+    return Base.print_array(io, matrix(mat))
 end
diff --git a/src/matrix_groups/eltary_matrices.jl b/src/matrix_groups/eltary_matrices.jl
index 42b582d..8181b11 100644
--- a/src/matrix_groups/eltary_matrices.jl
+++ b/src/matrix_groups/eltary_matrices.jl
@@ -2,23 +2,27 @@ 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(e::ElementaryMatrix{N,T}) where {N,T}
     m = StaticArrays.MMatrix{N,N,T}(LinearAlgebra.I)
diff --git a/src/matrix_groups/eltary_symplectic.jl b/src/matrix_groups/eltary_symplectic.jl
index 91ab941..1f70a5c 100644
--- 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
@@ -19,7 +24,7 @@ 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, "^$(s.val)")
+    return !isone(s.val) && print(io, "^$(s.val)")
 end
 
 _ind(s::ElementarySymplectic{N}) where {N} = (s.i, s.j)
@@ -41,21 +46,27 @@ 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(s::ElementarySymplectic{N,T}) where {N,T}
     @assert iseven(N)

From 3cdef72977f4b11cdb750c3915315c53df5c44f2 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 18:37:10 +0100
Subject: [PATCH 11/12] format aut_groups

---
 src/aut_groups/gersten_relations.jl | 30 +++++++++++++-----
 src/aut_groups/mcg.jl               | 49 ++++++++++++++++++++++-------
 src/aut_groups/sautFn.jl            | 16 +++++++---
 src/aut_groups/symplectic_twists.jl | 36 ++++++++++++++-------
 src/aut_groups/transvections.jl     | 21 ++++++++-----
 5 files changed, 109 insertions(+), 43 deletions(-)

diff --git a/src/aut_groups/gersten_relations.jl b/src/aut_groups/gersten_relations.jl
index d91a8ef..00e04f5 100644
--- 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
diff --git a/src/aut_groups/mcg.jl b/src/aut_groups/mcg.jl
index 1fc6286..cb10b96 100644
--- 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
diff --git a/src/aut_groups/sautFn.jl b/src/aut_groups/sautFn.jl
index b0805fa..f4fe6b4 100644
--- 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
diff --git a/src/aut_groups/symplectic_twists.jl b/src/aut_groups/symplectic_twists.jl
index 7a48233..a04ac08 100644
--- 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
diff --git a/src/aut_groups/transvections.jl b/src/aut_groups/transvections.jl
index 63c9717..04505d2 100644
--- 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
@@ -20,20 +20,23 @@ function Base.show(io::IO, t::Transvection)
         'λ'
     end
     print(io, id, subscriptify(t.i), '.', subscriptify(t.j))
-    t.inv && print(io, "^-1")
+    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 +87,7 @@ 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 +95,6 @@ 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

From af3913d0856d0e5ce80b0d4bfc246eed791ae632 Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Wed, 15 Mar 2023 19:07:14 +0100
Subject: [PATCH 12/12] more formatting

---
 .JuliaFormatter.toml                |  1 +
 src/Groups.jl                       |  9 +++-
 src/abelianize.jl                   | 12 +++--
 src/autgroups.jl                    | 47 +++++++++++++------
 src/constructions/direct_power.jl   | 58 +++++++++++++++--------
 src/constructions/direct_product.jl | 68 ++++++++++++++++++---------
 src/constructions/wreath_product.jl | 40 +++++++++-------
 src/hashing.jl                      |  8 +++-
 src/homomorphisms.jl                | 13 ++++--
 src/types.jl                        | 71 ++++++++++++++++-------------
 src/wl_ball.jl                      | 50 +++++++++++++-------
 11 files changed, 245 insertions(+), 132 deletions(-)
 create mode 120000 .JuliaFormatter.toml

diff --git a/.JuliaFormatter.toml b/.JuliaFormatter.toml
new file mode 120000
index 0000000..90107d8
--- /dev/null
+++ b/.JuliaFormatter.toml
@@ -0,0 +1 @@
+../.JuliaFormatter.toml
\ No newline at end of file
diff --git a/src/Groups.jl b/src/Groups.jl
index f34a7b7..96c7d09 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -14,7 +14,14 @@ 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
 
diff --git a/src/abelianize.jl b/src/abelianize.jl
index 1ab92ee..1ad1588 100644
--- 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)
@@ -50,7 +52,7 @@ 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(g) => word(g) for g in union(S, inv.(S)))
diff --git a/src/autgroups.jl b/src/autgroups.jl
index c255b47..2d4ff18 100644
--- a/src/autgroups.jl
+++ b/src/autgroups.jl
@@ -1,5 +1,5 @@
 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
@@ -26,7 +26,10 @@ 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)
@@ -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)
@@ -113,7 +130,11 @@ 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
 
@@ -135,13 +156,13 @@ function LettersMap(a::FPGroupElement{<:AutomorphismGroup})
     # (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))
+    indices_map =
+        Dict(A[A[fl]] => word(im) for (fl, im) in zip(first_letters, img))
     # inverses of generators are dealt lazily in getindex
 
     return LettersMap(indices_map, A)
 end
 
-
 function Base.getindex(lm::LettersMap, i::Integer)
     # here i is an index of an alphabet
     @boundscheck 1 ≤ i ≤ length(lm.A)
diff --git a/src/constructions/direct_power.jl b/src/constructions/direct_power.jl
index 25ffb4f..7e69b68 100644
--- 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}
@@ -49,8 +53,9 @@ 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)
 
@@ -83,13 +88,18 @@ 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))
+function Base.deepcopy_internal(g::DirectPowerElement, stackdict::IdDict)
+    return DirectPowerElement(
+        Base.deepcopy_internal(g.elts, stackdict),
+        parent(g),
+    )
+end
 
 Base.inv(g::DirectPowerElement) = DirectPowerElement(inv.(g.elts), parent(g))
 
@@ -98,15 +108,25 @@ 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)))
+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)
+    return print(io, "( ", join(g.elts, ", "), " )")
+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, ", "), " )")
diff --git a/src/constructions/direct_product.jl b/src/constructions/direct_product.jl
index 0ea6dfe..a522f6e 100644
--- 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}
@@ -50,15 +54,18 @@ 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
@@ -75,28 +82,45 @@ end
 
 GroupsCore.parent(g::DirectProductElement) = g.parent
 
-Base.:(==)(g::DirectProductElement, h::DirectProductElement) =
-    (parent(g) === parent(h) && g.elts == h.elts)
+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))
 
-Base.deepcopy_internal(g::DirectProductElement, stackdict::IdDict) =
-    DirectProductElement(Base.deepcopy_internal(g.elts, stackdict), parent(g))
+function Base.deepcopy_internal(g::DirectProductElement, stackdict::IdDict)
+    return DirectProductElement(
+        Base.deepcopy_internal(g.elts, stackdict),
+        parent(g),
+    )
+end
 
-Base.inv(g::DirectProductElement) =
-    DirectProductElement(inv.(g.elts), parent(g))
+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
 
-GroupsCore.order(::Type{I}, g::DirectProductElement) where {I<:Integer} =
-    convert(I, lcm(order(I, first(g.elts)), order(I, last(g.elts))))
+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
 
 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)
+    return print(io, "( $(join(g.elts, ",")) )")
+end
+
+# convienience:
+Base.@propagate_inbounds function Base.getindex(
+    g::DirectProductElement,
+    i::Integer,
+)
+    return g.elts[i]
+end
diff --git a/src/constructions/wreath_product.jl b/src/constructions/wreath_product.jl
index aa61198..bb4fa57 100644
--- a/src/constructions/wreath_product.jl
+++ b/src/constructions/wreath_product.jl
@@ -1,4 +1,5 @@
-import PermutationGroups: AbstractPermutationGroup, AbstractPerm, degree, SymmetricGroup
+import PermutationGroups:
+    AbstractPermutationGroup, AbstractPerm, degree, SymmetricGroup
 
 """
     WreathProduct(G::Group, P::AbstractPermutationGroup) <: Group
@@ -38,21 +39,22 @@ struct WreathProductElement{
         p::AbstractPerm,
         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))
+    return elt, (iterator = itr, state = last(res))
 end
 
 function Base.iterate(G::WreathProduct, state)
@@ -60,7 +62,7 @@ function Base.iterate(G::WreathProduct, state)
     res = iterate(itr, st)
     res === nothing && return nothing
     elt = WreathProductElement(first(res)..., G)
-    return elt, (iterator=itr, state=last(res))
+    return elt, (iterator = itr, state = last(res))
 end
 
 function Base.IteratorSize(::Type{<:WreathProduct{DP,PGr}}) where {DP,PGr}
@@ -78,8 +80,9 @@ 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)]
@@ -93,18 +96,19 @@ 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
+function Base.:(==)(g::WreathProductElement, h::WreathProductElement)
+    return parent(g) === parent(h) && g.n == h.n && g.p == h.p
+end
 
-Base.hash(g::WreathProductElement, h::UInt) =
-    hash(g.n, hash(g.p, hash(g.parent, h)))
+function Base.hash(g::WreathProductElement, h::UInt)
+    return hash(g.n, hash(g.p, hash(g.parent, h)))
+end
 
 function Base.deepcopy_internal(g::WreathProductElement, stackdict::IdDict)
     return WreathProductElement(
@@ -114,8 +118,9 @@ function Base.deepcopy_internal(g::WreathProductElement, stackdict::IdDict)
     )
 end
 
-_act(p::AbstractPerm, n::DirectPowerElement) =
-    DirectPowerElement(n.elts^p, parent(n))
+function _act(p::AbstractPerm, n::DirectPowerElement)
+    return DirectPowerElement(n.elts^p, parent(n))
+end
 
 function Base.inv(g::WreathProductElement)
     pinv = inv(g.p)
@@ -129,8 +134,9 @@ 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)")
+function Base.show(io::IO, G::WreathProduct)
+    return print(io, "Wreath product of $(G.N.group) by $(G.P)")
+end
 Base.show(io::IO, g::WreathProductElement) = print(io, "( $(g.n)≀$(g.p) )")
 
 Base.copy(g::WreathProductElement) = WreathProductElement(g.n, g.p, parent(g))
diff --git a/src/hashing.jl b/src/hashing.jl
index ed729b1..5c33019 100644
--- a/src/hashing.jl
+++ b/src/hashing.jl
@@ -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`:
diff --git a/src/homomorphisms.jl b/src/homomorphisms.jl
index d045d47..09e9355 100644
--- 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
diff --git a/src/types.jl b/src/types.jl
index bbe332f..336b570 100644
--- 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")
 
@@ -65,8 +63,9 @@ 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)]
+function GroupsCore.gens(G::AbstractFPGroup)
+    return [gens(G, i) for i in 1:GroupsCore.ngens(G)]
+end
 
 # TODO: ProductReplacementAlgorithm
 function Base.rand(
@@ -79,9 +78,11 @@ function Base.rand(
     return FPGroupElement(word_type(G)(rand(1:nletters, l)), 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
 
@@ -93,18 +94,22 @@ mutable struct FPGroupElement{Gr<:AbstractFPGroup,W<:AbstractWord} <:
     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, "{$Gr, …}")
+function Base.show(io::IO, ::Type{<:FPGroupElement{Gr}}) where {Gr}
+    return print(io, FPGroupElement, "{$Gr, …}")
+end
 
 word(f::AbstractFPGroupElement) = f.word
 
@@ -142,11 +147,12 @@ function Base.:(*)(g::GEl, h::GEl) where {GEl<:AbstractFPGroupElement}
     return GEl(word(g) * word(h), parent(g))
 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)))
@@ -167,9 +173,7 @@ end
 FreeGroup(gens, A::Alphabet) = FreeGroup(gens, KnuthBendix.LenLex(A))
 
 function FreeGroup(A::Alphabet)
-    @boundscheck @assert all(
-        KnuthBendix.hasinverse(l, A) for l in A
-    )
+    @boundscheck @assert all(KnuthBendix.hasinverse(l, A) for l in A)
     gens = Vector{eltype(A)}()
     invs = Vector{eltype(A)}()
     for l in A
@@ -193,8 +197,9 @@ function FreeGroup(n::Integer)
     return FreeGroup(symbols[1:2:2n], Alphabet(symbols, inverses))
 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 +210,9 @@ 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 +228,8 @@ 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
@@ -253,8 +259,9 @@ 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
 
diff --git a/src/wl_ball.jl b/src/wl_ball.jl
index 0c52080..ae8db7d 100644
--- a/src/wl_ball.jl
+++ b/src/wl_ball.jl
@@ -8,22 +8,40 @@ 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 = *,
+    threading = true,
 ) where {T}
-    threading && return wlmetric_ball_thr(S, center, radius=radius, op=op)
-    return wlmetric_ball_serial(S, center, radius=radius, op=op)
+    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}
+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!)
+    old = union!(OrderedSet([center]), [center * s for s in S])
+    sizes = [1, length(old)]
+    for _ in 2:radius
+        new = collect(
+            op(o, s) for o in @view(old.dict.keys[sizes[end-1]:end]) for s in S
+        )
+        union!(old, new)
+        push!(sizes, length(old))
+    end
+    return old.dict.keys, sizes[2:end]
 end
 
-function wlmetric_ball_thr(S::AbstractVector{T}, center::T=one(first(S)); radius=2, op=*) where {T}
+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)
@@ -31,12 +49,13 @@ 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,
+    for _ in 2:radius
+        old = let old = old, S = S
             new = collect(
-                (g = op(o, s); normalform!(g); hash(g); g)
-                # normalform! and hash are to make assure thread-safety of produced elts
-                for o in @view(old[sizes[end-1]:end]) for s in S
+                (g = op(o, s);
+                normalform!(g);
+                hash(g);
+                g) for o in @view(old[sizes[end-1]:end]) for s in S
             )
 
             append!(old, new)
@@ -46,4 +65,3 @@ function _wlmetric_ball(S, old, radius, op, collect, unique)
     end
     return old, sizes[2:end]
 end
-