diff --git a/src/AutGroup.jl b/src/AutGroup.jl
index 1c9206a..2fd2185 100644
--- a/src/AutGroup.jl
+++ b/src/AutGroup.jl
@@ -297,24 +297,28 @@ function reduce!(w::Automorphism)
     return W
 end
 
-function linear_repr(A::Automorphism{N}, hom=matrix_repr) where N
-    return reduce(*, linear_repr.(A.symbols, N, hom), init=hom(Identity(),N,1))
-end
+###############################################################################
+#
+#   Abelianization (natural Representation to GL(N,Z))
+#
 
-linear_repr(a::AutSymbol, n::Int, hom) = hom(a.fn, n, a.pow)
+abelianize(A::Automorphism{N}) where N = image(A, abelianize; n=N)
 
-function matrix_repr(a::Union{RTransvect, LTransvect}, n::Int, pow)
+# homomorphism definition
+abelianize(; n::Integer=1) = Matrix{Int}(I, n, n)
+abelianize(a::AutSymbol; n::Int=1) = abelianize(a.fn, n, a.pow)
+
+function abelianize(a::Union{RTransvect, LTransvect}, n::Int, pow)
     x = Matrix{Int}(I, n, n)
     x[a.i,a.j] = pow
     return x
 end
 
-function matrix_repr(a::FlipAut, n::Int, pow)
+function abelianize(a::FlipAut, n::Int, pow)
     x = Matrix{Int}(I, n, n)
-    x[a.i,a.i] = -1^pow
+    x[a.i,a.i] = -1
     return x
 end
 
-matrix_repr(a::PermAut, n::Int, pow) = Matrix{Int}(I, n, n)[(a.perm^pow).d, :]
-
-matrix_repr(a::Identity, n::Int, pow) = Matrix{Int}(I, n, n)
+abelianize(a::PermAut, n::Integer, pow) = Matrix{Int}(I, n, n)[(a.perm^pow).d, :]
+abelianize(a::Identity, n::Integer, pow) = abelianize(;n=n)
diff --git a/src/Groups.jl b/src/Groups.jl
index 443dd46..9dc67cb 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -13,6 +13,7 @@ import Base: deepcopy_internal
 using LinearAlgebra
 using Markdown
 
+export gens, FreeGroup, Aut, SAut
 
 include("types.jl")
 
@@ -89,4 +90,17 @@ function generate_balls(S::AbstractVector{T}, center::T=one(first(S));
     return c.*B, sizes
 end
 
+@doc doc"""
+    image(A::GWord, homomorphism; kwargs...)
+Evaluate homomorphism `homomorphism` on a GWord `A`.
+`homomorphism` needs implement
+  > `hom(s; kwargs...)`,
+where `hom(;kwargs...)` evaluates the value at the identity element.
+"""
+function image(w::GWord, hom; kwargs...)
+    return reduce(*,
+        (hom(s; kwargs...) for s in syllables(w)),
+        init = hom(;kwargs...))
+end
+
 end # of module Groups
diff --git a/test/AutGroup-tests.jl b/test/AutGroup-tests.jl
index 8699874..b4e5a63 100644
--- a/test/AutGroup-tests.jl
+++ b/test/AutGroup-tests.jl
@@ -9,7 +9,7 @@
       @test isa(f, Groups.GSymbol)
       @test isa(f, Groups.AutSymbol)
       @test isa(Groups.AutSymbol(perm"(4)"), Groups.AutSymbol)
-      @test isa(Groups.AutSymbol([2,3,4,1]), Groups.AutSymbol)
+      @test isa(Groups.AutSymbol(perm"(1,2,3,4)"), Groups.AutSymbol)
       @test isa(Groups.transvection_R(1,2), Groups.AutSymbol)
       @test isa(Groups.transvection_R(3,4), Groups.AutSymbol)
       @test isa(Groups.flip(3), Groups.AutSymbol)
@@ -205,48 +205,48 @@
       @test length(unique(B_2)) == 1777
    end
 
-   @testset "linear_repr tests" begin
-      N = 3
+   @testset "abelianization homomorphism" begin
+      N = 4
       G = AutGroup(FreeGroup(N))
       S = unique([gens(G); inv.(gens(G))])
       R = 3
 
-      @test Groups.linear_repr(one(G)) isa Matrix{Int}
-      @test Groups.linear_repr(one(G)) == Matrix{Int}(I, N, N)
+      @test Groups.abelianize(one(G)) isa Matrix{Int}
+      @test Groups.abelianize(one(G)) == Matrix{Int}(I, N, N)
 
       M = Matrix{Int}(I, N, N)
       M[1,2] = 1
-      ϱ₁₂ = G(Groups.transvection_R(1,2))
-      λ₁₂ = G(Groups.transvection_R(1,2))
+      ϱ₁₂ = G(Groups.ϱ(1,2))
+      λ₁₂ = G(Groups.λ(1,2))
 
-      @test Groups.linear_repr(ϱ₁₂) == M
-      @test Groups.linear_repr(λ₁₂) == M
+      @test Groups.abelianize(ϱ₁₂) == M
+      @test Groups.abelianize(λ₁₂) == M
 
       M[1,2] = -1
 
-      @test Groups.linear_repr(ϱ₁₂^-1) == M
-      @test Groups.linear_repr(λ₁₂^-1) == M
+      @test Groups.abelianize(ϱ₁₂^-1) == M
+      @test Groups.abelianize(λ₁₂^-1) == M
 
-      @test Groups.linear_repr(ϱ₁₂*λ₁₂^-1) == Matrix{Int}(I, N, N)
-      @test Groups.linear_repr(λ₁₂^-1*ϱ₁₂) == Matrix{Int}(I, N, N)
+      @test Groups.abelianize(ϱ₁₂*λ₁₂^-1) == Matrix{Int}(I, N, N)
+      @test Groups.abelianize(λ₁₂^-1*ϱ₁₂) == Matrix{Int}(I, N, N)
 
       M = Matrix{Int}(I, N, N)
       M[2,2] = -1
       ε₂ = G(Groups.flip(2))
 
-      @test Groups.linear_repr(ε₂) == M
-      @test Groups.linear_repr(ε₂^2) == Matrix{Int}(I, N, N)
+      @test Groups.abelianize(ε₂) == M
+      @test Groups.abelianize(ε₂^2) == Matrix{Int}(I, N, N)
 
-      M = [0 1 0; 0 0 1; 1 0 0]
+      M = [0 1 0 0; 0 0 0 1; 0 0 1 0; 1 0 0 0]
 
-      σ = G(Groups.AutSymbol(perm"(1,2,3)"))
-      @test Groups.linear_repr(σ) == M
-      @test Groups.linear_repr(σ^3) == Matrix{Int}(I, 3, 3)
-      @test Groups.linear_repr(σ)^3 == Matrix{Int}(I, 3, 3)
+      σ = G(Groups.AutSymbol(perm"(1,2,4)"))
+      @test Groups.abelianize(σ) == M
+      @test Groups.abelianize(σ^3) == Matrix{Int}(I, N, N)
+      @test Groups.abelianize(σ)^3 == Matrix{Int}(I, N, N)
 
       function test_homomorphism(S, r)
          for elts in Iterators.product([[g for g in S] for _ in 1:r]...)
-            prod(Groups.linear_repr.(elts)) == Groups.linear_repr(prod(elts)) || error("linear representaton test failed at $elts")
+            prod(Groups.abelianize.(elts)) == Groups.abelianize(prod(elts)) || error("linear representaton test failed at $elts")
          end
          return 0
       end