diff --git a/src/AutGroup.jl b/src/AutGroup.jl
index dcbb1e1..fa05877 100644
--- a/src/AutGroup.jl
+++ b/src/AutGroup.jl
@@ -91,7 +91,7 @@ function change_pow(s::AutSymbol, n::Integer)
     elseif symbol isa LTransvect
         return transvection_L(symbol.i, symbol.j, n)
     elseif symbol isa Identity
-        return s
+        return id_autsymbol()
     else
         throw(DomainError("Unknown type of AutSymbol: $s"))
     end
diff --git a/src/arithmetic.jl b/src/arithmetic.jl
index c7e8c21..f84527d 100644
--- a/src/arithmetic.jl
+++ b/src/arithmetic.jl
@@ -1,5 +1,5 @@
 function Base.inv(W::T) where T<:GWord
-    length(W) == 0 && return W
+    length(W) == 0 && return one(W)
     G = parent(W)
     w = T([inv(s) for s in Iterators.reverse(syllables(W))])
     return setparent!(w, G)
diff --git a/test/AutGroup-tests.jl b/test/AutGroup-tests.jl
index 50621e2..0b9c658 100644
--- a/test/AutGroup-tests.jl
+++ b/test/AutGroup-tests.jl
@@ -16,7 +16,8 @@
 
       @test Groups.id_autsymbol() isa Groups.AutSymbol
       @test inv(Groups.id_autsymbol()) isa Groups.AutSymbol
-      @test inv(Groups.id_autsymbol()) == Groups.id_autsymbol()
+      x = Groups.id_autsymbol()
+      @test inv(x) == Groups.id_autsymbol()
    end
 
    a,b,c,d = gens(FreeGroup(4))
@@ -266,7 +267,7 @@
       @test Groups.abelianize(σ^3) == Matrix{Int}(I, N, N)
       @test Groups.abelianize(σ)^3 == Matrix{Int}(I, N, N)
 
-      @test Groups.abelianize(G(Groups.id_autsymbol())) == Matrix{Int}(I, N, N) 
+      @test Groups.abelianize(G(Groups.id_autsymbol())) == Matrix{Int}(I, N, N)
 
       function test_homomorphism(S, r)
          for elts in Iterators.product([[g for g in S] for _ in 1:r]...)
diff --git a/test/FreeGroup-tests.jl b/test/FreeGroup-tests.jl
index 3a1863f..49f4b82 100644
--- a/test/FreeGroup-tests.jl
+++ b/test/FreeGroup-tests.jl
@@ -109,6 +109,13 @@ end
       tt = deepcopy(t)
       append!(tt, s, inv(t))
       @test string(tt) == "t*s*t^-1"
+
+      o = one(t)
+      o_inv = inv(o)
+      @test o == o_inv
+      @test o !== o_inv
+      Groups.rmul!(o, t)
+      @test o != o_inv
    end
 
    @testset "reductions" begin