diff --git a/Project.toml b/Project.toml
index c89da1c..7de2f34 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 = "0.7.8"
+version = "0.8"
 
 [deps]
 GroupsCore = "d5909c97-4eac-4ecc-a3dc-fdd0858a4120"
@@ -14,10 +14,10 @@ Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 
 [compat]
-GroupsCore = "0.4"
+GroupsCore = "0.5"
 KnuthBendix = "0.4"
 OrderedCollections = "1"
-PermutationGroups = "0.4"
+PermutationGroups = "0.6"
 StaticArrays = "1"
 julia = "1.6"
 
diff --git a/src/Groups.jl b/src/Groups.jl
index 9237429..a364c6f 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -27,6 +27,7 @@ include(joinpath("constructions", "constructions.jl"))
 import .Constructions
 
 include("types.jl")
+include("rand.jl")
 include("hashing.jl")
 include("normalform.jl")
 include("autgroups.jl")
diff --git a/src/autgroups.jl b/src/autgroups.jl
index 75b8eba..f361b37 100644
--- a/src/autgroups.jl
+++ b/src/autgroups.jl
@@ -36,8 +36,8 @@ function Base.:(==)(
         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
 
@@ -138,43 +138,47 @@ 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
@@ -185,9 +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
diff --git a/src/constructions/direct_power.jl b/src/constructions/direct_power.jl
index 7e69b68..3bb3114 100644
--- a/src/constructions/direct_power.jl
+++ b/src/constructions/direct_power.jl
@@ -59,15 +59,6 @@ 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)
@@ -78,14 +69,6 @@ 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
 
 function Base.:(==)(g::DirectPowerElement, h::DirectPowerElement)
@@ -94,13 +77,6 @@ end
 
 Base.hash(g::DirectPowerElement, h::UInt) = hash(g.elts, hash(parent(g), h))
 
-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))
 
 function Base.:(*)(g::DirectPowerElement, h::DirectPowerElement)
@@ -108,6 +84,39 @@ function Base.:(*)(g::DirectPowerElement, h::DirectPowerElement)
     return DirectPowerElement(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::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
@@ -117,10 +126,13 @@ 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"
-    return 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, ", "), " )")
+    print(io, "( ")
+    join(io, g.elts, ", ")
+    return print(" )")
 end
 
 # convienience:
diff --git a/src/constructions/direct_product.jl b/src/constructions/direct_product.jl
index a522f6e..adfe0b7 100644
--- a/src/constructions/direct_product.jl
+++ b/src/constructions/direct_product.jl
@@ -72,14 +72,6 @@ end
 
 Base.isfinite(G::DirectProduct) = isfinite(G.first) && isfinite(G.last)
 
-function Base.rand(
-    rng::Random.AbstractRNG,
-    rs::Random.SamplerTrivial{<:DirectProduct},
-)
-    G = rs[]
-    return DirectProductElement((rand(rng, G.first), rand(rng, G.last)), G)
-end
-
 GroupsCore.parent(g::DirectProductElement) = g.parent
 
 function Base.:(==)(g::DirectProductElement, h::DirectProductElement)
@@ -88,13 +80,6 @@ end
 
 Base.hash(g::DirectProductElement, h::UInt) = hash(g.elts, hash(parent(g), h))
 
-function Base.deepcopy_internal(g::DirectProductElement, stackdict::IdDict)
-    return DirectProductElement(
-        Base.deepcopy_internal(g.elts, stackdict),
-        parent(g),
-    )
-end
-
 function Base.inv(g::DirectProductElement)
     return DirectProductElement(inv.(g.elts), parent(g))
 end
@@ -104,6 +89,30 @@ function Base.:(*)(g::DirectProductElement, h::DirectProductElement)
     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},
+)
+    G = rs[]
+    return DirectProductElement((rand(rng, G.first), rand(rng, G.last)), G)
+end
+
 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
@@ -111,10 +120,13 @@ end
 Base.isone(g::DirectProductElement) = all(isone, g.elts)
 
 function Base.show(io::IO, G::DirectProduct)
-    return print(io, "Direct product of $(G.first) and $(G.last)")
+    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, ",")) )")
+    print(io, "( ")
+    join(io, g.elts, ", ")
+    return print(io, " )")
 end
 
 # convienience:
diff --git a/src/constructions/wreath_product.jl b/src/constructions/wreath_product.jl
index cf0608a..2f99208 100644
--- a/src/constructions/wreath_product.jl
+++ b/src/constructions/wreath_product.jl
@@ -1,7 +1,4 @@
-import PermutationGroups:
-    AbstractPermutationGroup,
-    AbstractPermutation,
-    degree
+import PermutationGroups as PG
 
 """
     WreathProduct(G::Group, P::AbstractPermutationGroup) <: Group
@@ -16,20 +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<:AbstractPermutation,
+    PEl<:PG.AP.AbstractPermutation,
     Wr<:WreathProduct,
 } <: GroupsCore.GroupElement
     n::DPEl
@@ -38,7 +35,7 @@ struct WreathProductElement{
 
     function WreathProductElement(
         n::DirectPowerElement,
-        p::AbstractPermutation,
+        p::PG.AP.AbstractPermutation,
         W::WreathProduct,
     )
         return new{typeof(n),typeof(p),typeof(W)}(n, p, W)
@@ -97,14 +94,6 @@ 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
 
 function Base.:(==)(g::WreathProductElement, h::WreathProductElement)
@@ -115,15 +104,7 @@ 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(
-        Base.deepcopy_internal(g.n, stackdict),
-        Base.deepcopy_internal(g.p, stackdict),
-        parent(g),
-    )
-end
-
-function _act(p::AbstractPermutation, n::DirectPowerElement)
+function _act(p::PG.AP.AbstractPermutation, n::DirectPowerElement)
     return DirectPowerElement(
         ntuple(i -> n.elts[i^p], length(n.elts)),
         parent(n),
@@ -140,11 +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)
 
 function Base.show(io::IO, G::WreathProduct)
-    return print(io, "Wreath product of $(G.N.group) by $(G.P)")
+    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))
+function Base.show(io::IO, g::WreathProductElement)
+    return print(io, "( ", g.n, "≀", g.p, " )")
+end
diff --git a/src/normalform.jl b/src/normalform.jl
index 7b7742c..613e51a 100644
--- 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))
diff --git a/src/rand.jl b/src/rand.jl
new file mode 100644
index 0000000..1ead576
--- /dev/null
+++ 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
diff --git a/src/types.jl b/src/types.jl
index d6a567e..3b7c150 100644
--- a/src/types.jl
+++ b/src/types.jl
@@ -67,17 +67,6 @@ function GroupsCore.gens(G::AbstractFPGroup)
     return [gens(G, i) for i in 1:GroupsCore.ngens(G)]
 end
 
-# 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)
-end
-
 function Base.isfinite(::AbstractFPGroup)
     return (
         @warn "using generic isfinite(::AbstractFPGroup): the returned `false` might be wrong"; false
@@ -110,10 +99,6 @@ mutable struct FPGroupElement{Gr<:AbstractFPGroup,W<:AbstractWord} <:
     end
 end
 
-function Base.show(io::IO, ::Type{<:FPGroupElement{Gr}}) where {Gr}
-    return print(io, FPGroupElement, "{$Gr, …}")
-end
-
 function Base.copy(f::FPGroupElement)
     return FPGroupElement(copy(word(f)), parent(f), f.savedhash)
 end
@@ -134,18 +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)
-    haskey(stackdict, objectid(g)) && return stackdict[objectid(g)]
-    cw = if haskey(stackdict, objectid(word(g)))
-        stackdict[objectid(word(g))]
-    else
-        copy(word(g))
-    end
-    return FPGroupElement(cw, 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}
@@ -192,9 +192,16 @@ 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)
+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)}()
@@ -203,20 +210,12 @@ 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
 
 function Base.show(io::IO, F::FreeGroup)
@@ -265,10 +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
 
diff --git a/test/AutFn.jl b/test/AutFn.jl
index c3f035b..85b1837 100644
--- 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)
 
diff --git a/test/fp_groups.jl b/test/fp_groups.jl
index 87e6ef7..47511a6 100644
--- 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)
diff --git a/test/free_groups.jl b/test/free_groups.jl
index f35d85e..9604a1d 100644
--- 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