diff --git a/src/DirectPower.jl b/src/DirectPower.jl
index ec0140f..fe87840 100644
--- a/src/DirectPower.jl
+++ b/src/DirectPower.jl
@@ -65,7 +65,7 @@ parent(g::DirectPowerGroupElem{N, T}) where {N,T} =
 #
 ###############################################################################
 
-size(g::DirectPowerGroupElem{N}) where N = (N,)
+Base.size(g::DirectPowerGroupElem{N}) where N = (N,)
 Base.IndexStyle(::Type{DirectPowerGroupElem}) = Base.LinearFast()
 Base.getindex(g::DirectPowerGroupElem, i::Int) = g.elts[i]
 
@@ -188,7 +188,7 @@ end
 ###############################################################################
 
 order(G::DirectPowerGroup{N}) where N = order(G.group)^N
-length(G::DirectPowerGroup) = order(G)
+Base.length(G::DirectPowerGroup) = order(G)
 
 function iterate(G::DirectPowerGroup{N}) where N
    elts = collect(G.group)
diff --git a/src/symbols.jl b/src/symbols.jl
index dd766ff..e13435c 100644
--- a/src/symbols.jl
+++ b/src/symbols.jl
@@ -3,8 +3,9 @@ change_pow(s::S, n::Integer) where S<:GSymbol = S(s.id, n)
 function Base.iterate(s::GS, i=1) where GS<:GSymbol
     return i <= abs(s.pow) ? (GS(s.id, sign(s.pow)), i+1) : nothing
 end
-Base.length(s::GSymbol) = abs(s.pow)
-Base.size(s::GSymbol) = (length(s), )
+Base.size(s::GSymbol) = (abs(s.pow), )
+Base.length(s::GSymbol) = first(size(s))
+
 Base.eltype(s::GS) where GS<:GSymbol = GS
 
 Base.isone(s::GSymbol) = iszero(s.pow)
diff --git a/test/FreeGroup-tests.jl b/test/FreeGroup-tests.jl
index fae5c6d..3a1863f 100644
--- a/test/FreeGroup-tests.jl
+++ b/test/FreeGroup-tests.jl
@@ -38,6 +38,7 @@ end
    @test length(FreeGroupElem(t)) == 2
    @test Groups.FreeSymbol(:s, 1) != Groups.FreeSymbol(:s, 2)
    @test Groups.FreeSymbol(:s, 1) != Groups.FreeSymbol(:t, 1)
+   @test collect(Groups.FreeSymbol(:s, 2)) == [i for i in Groups.FreeSymbol(:s, 2)] == [s, s]
 end
 
 @testset "FreeGroup" begin
@@ -52,8 +53,10 @@ end
       @test eltype(gens(G)) == FreeGroupElem
       @test length(gens(G)) == 2
 
-      @test collect(s*t) == Groups.syllables(s*t)
       tt, ss = Groups.FreeSymbol(:t), Groups.FreeSymbol(:s)
+      @test Groups.GroupWord([tt, inv(tt)]) isa FreeGroupElem
+
+      @test collect(s*t) == Groups.syllables(s*t)
       @test collect(t^2) == [tt, tt]
       ttinv = Groups.FreeSymbol(:t, -1)
       w = t^-2*s^3*t^2
@@ -66,7 +69,8 @@ end
 
       @test collect(ttinv) == [ttinv]
 
-      @test Groups.GroupWord([tt, inv(tt)]) isa FreeGroupElem
+      @test isone(t^0)
+      @test !isone(t^1)
    end
 
    @testset "internal arithmetic" begin
@@ -80,6 +84,13 @@ end
       tt = deepcopy(t)
       @test string(Groups.lmul!(tt, tt, inv(tt))) == "(id)"
 
+      w = deepcopy(t)
+      @test length(Groups.rmul!(w, t)) == 2
+      @test length(Groups.lmul!(w, inv(t))) == 1
+      w = AbstractAlgebra.mul!(w, w, s)
+      @test length(w) == 2
+      @test length(Groups.lmul!(w, inv(s))) == 3
+
       tt = deepcopy(t)
       push!(tt, inv(t_symb))
       @test string(tt) == "t*t^-1"
@@ -150,7 +161,7 @@ end
       @test findlast(s^-1*t^-1, c) == 3
       @test findprev(s, s*t*s*t, 4) == 3
       @test findprev(s*t, s*t*s*t, 2) == 1
-      @test findlast(t^2*s, c) === nothing
+      @test findprev(Groups.FreeSymbol(:t, 2), c, 4) === nothing
 
       w = s*t*s^-1
       subst = Dict{FreeGroupElem, FreeGroupElem}(w => s^1, s*t^-1 => t^4)