diff --git a/src/Groups.jl b/src/Groups.jl
index 949efd2..0934ddb 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -20,6 +20,7 @@ include("fallbacks.jl")
 include("words.jl")
 include("hashing.jl")
 include("freereduce.jl")
+include("arithmetic.jl")
 
 include("FreeGroup.jl")
 include("FPGroups.jl")
@@ -62,102 +63,6 @@ function show(io::IO, s::T) where {T<:GSymbol}
       print(io, string((s.id))*"^$(s.pow)")
    end
 end
-
-###############################################################################
-#
-#   Binary operators
-#
-###############################################################################
-
-function Base.append!(w::GWord{T}, v::AbstractVector{T}) where T
-    append!(syllables(w), v)
-    return w
-end
-
-function Base.prepend!(w::GWord{T}, v::AbstractVector{T}) where T
-    prepend!(syllables(w), v)
-    return w
-end
-
-Base.append!(w::T, v::T) where T <: GWord = append!(w, syllables(v))
-Base.prepend!(w::T, v::T) where T <: GWord = prepend!(w, syllables(v))
-
-for (mul, f) in ((:rmul!, :push!), (:lmul!, :pushfirst!))
-    @eval begin
-        function $mul(out::T, w::T, s::GSymbol) where T <:GWord
-            $f(syllables(out), s)
-            return freereduce!(out)
-        end
-    end
-end
-
-function rmul!(out::T, x::T, y::T) where T<: GWord
-    if out === x
-        out = deepcopy(out)
-        return freereduce!(append!(out, y))
-    elseif out === y
-        out = deepcopy(out)
-        return freereduce!(prepend!(out, x))
-    else
-        slenx = syllablelength(x)
-        sleny = syllablelength(y)
-        resize!(syllables(out), slenx+sleny)
-        syllables(out)[1:slenx] .= syllables(x)
-        syllables(out)[slenx+1:slenx+sleny] .= syllables(y)
-        return freereduce!(out)
-    end
-end
-
-lmul!(out::T, x::T, y::T) where T <: GWord = rmul!(out, y, x)
-
-function AbstractAlgebra.mul!(out::T, x::T, y::T) where T <: GWord
-    return rmul!(out, x, y)
-end
-
-(*)(W::GW, Z::GW) where GW <: GWord = rmul!(deepcopy(W), W, Z)
-(*)(W::GWord, s::GSymbol) = rmul!(deepcopy(W), W, s)
-(*)(s::GSymbol, W::GWord) = lmul!(deepcopy(W), W, s)
-
-function power_by_squaring(W::GWord, p::Integer)
-    if p < 0
-        return power_by_squaring(inv(W), -p)
-    elseif p == 0
-        return one(parent(W))
-    elseif p == 1
-        return W
-    elseif p == 2
-        return W*W
-    end
-    W = deepcopy(W)
-    t = trailing_zeros(p) + 1
-    p >>= t
-    while (t -= 1) > 0
-        append!(W, W)
-    end
-    Z = deepcopy(W)
-    while p > 0
-        t = trailing_zeros(p) + 1
-        p >>= t
-        while (t -= 1) >= 0
-            append!(W, W)
-        end
-        append!(Z, W)
-    end
-
-    return freereduce!(Z)
-end
-
-(^)(x::GWord, n::Integer) = power_by_squaring(x,n)
-
-###############################################################################
-#
-#   Inversion
-#
-###############################################################################
-
-function inv(W::T) where T<:GWord
-    if length(W) == 0
-        return W
     else
         G = parent(W)
         w = T([inv(s) for s in Iterators.reverse(syllables(W))])
diff --git a/src/arithmetic.jl b/src/arithmetic.jl
new file mode 100644
index 0000000..933c5ec
--- /dev/null
+++ b/src/arithmetic.jl
@@ -0,0 +1,93 @@
+function Base.inv(W::T) where T<:GWord
+    length(W) == 0 && return W
+    G = parent(W)
+    w = T([inv(s) for s in Iterators.reverse(syllables(W))])
+    return setparent!(w, G)
+end
+
+###############################################################################
+#
+#   Binary operators
+#
+
+function Base.append!(w::GWord{T}, v::AbstractVector{T}) where T
+    append!(syllables(w), v)
+    return w
+end
+
+function Base.prepend!(w::GWord{T}, v::AbstractVector{T}) where T
+    prepend!(syllables(w), v)
+    return w
+end
+
+Base.append!(w::T, v::T) where T <: GWord = append!(w, syllables(v))
+Base.prepend!(w::T, v::T) where T <: GWord = prepend!(w, syllables(v))
+
+for (mul, f) in ((:rmul!, :push!), (:lmul!, :pushfirst!))
+    @eval begin
+        function $mul(out::T, w::T, s::GSymbol) where T <:GWord
+            resize!(syllables(out), syllablelength(w))
+            syllables(out) .= syllables(w)
+            $f(syllables(out), s)
+            return freereduce!(out)
+        end
+    end
+end
+
+function rmul!(out::T, x::T, y::T) where T<: GWord
+    if out === x
+        out = deepcopy(out)
+        return freereduce!(append!(out, y))
+    elseif out === y
+        out = deepcopy(out)
+        return freereduce!(prepend!(out, x))
+    else
+        slenx = syllablelength(x)
+        sleny = syllablelength(y)
+        resize!(syllables(out), slenx+sleny)
+        syllables(out)[1:slenx] .= syllables(x)
+        syllables(out)[slenx+1:slenx+sleny] .= syllables(y)
+        return freereduce!(out)
+    end
+end
+
+lmul!(out::T, x::T, y::T) where T <: GWord = rmul!(out, y, x)
+
+function AbstractAlgebra.mul!(out::T, x::T, y::T) where T <: GWord
+    return rmul!(out, x, y)
+end
+
+(*)(W::GW, Z::GW) where GW <: GWord = rmul!(deepcopy(W), W, Z)
+(*)(W::GWord, s::GSymbol) = rmul!(deepcopy(W), W, s)
+(*)(s::GSymbol, W::GWord) = lmul!(deepcopy(W), W, s)
+
+function power_by_squaring(W::GWord, p::Integer)
+    if p < 0
+        return power_by_squaring(inv(W), -p)
+    elseif p == 0
+        return one(W)
+    elseif p == 1
+        return W
+    elseif p == 2
+        return W*W
+    end
+    W = deepcopy(W)
+    t = trailing_zeros(p) + 1
+    p >>= t
+    while (t -= 1) > 0
+        append!(W, W)
+    end
+    Z = deepcopy(W)
+    while p > 0
+        t = trailing_zeros(p) + 1
+        p >>= t
+        while (t -= 1) >= 0
+            append!(W, W)
+        end
+        append!(Z, W)
+    end
+
+    return freereduce!(Z)
+end
+
+(^)(x::GWord, n::Integer) = power_by_squaring(x,n)