diff --git a/test/AutGroup-tests.jl b/test/AutGroup-tests.jl
index 7457c57..48d9cb8 100644
--- a/test/AutGroup-tests.jl
+++ b/test/AutGroup-tests.jl
@@ -70,7 +70,6 @@
       @test      l(deepcopy(D)) == (a, b, c, c*d)
       @test inv(l)(deepcopy(D)) == (a, b, c, c^-1*d)
 
-
       i,j = 2,4
       r = Groups.rmul_autsymbol(i,j)
       l = Groups.lmul_autsymbol(i,j)
@@ -81,19 +80,20 @@
    end
 
    @testset "AutGroup/Automorphism constructors" begin
+
       f = Groups.AutSymbol("a", 1, Groups.FlipAut(1))
       @test isa(Automorphism{3}(f), Groups.GWord)
       @test isa(Automorphism{3}(f), Automorphism)
-      @test isa(AutGroup(FreeGroup(3)), Group)
+      @test isa(AutGroup(FreeGroup(3)), AbstractAlgebra.Group)
       @test isa(AutGroup(FreeGroup(1)), Groups.AbstractFPGroup)
       A = AutGroup(FreeGroup(1))
-      @test isa(gens(A), Vector{Automorphism{1}})
-      @test length(gens(A)) == 1
+      @test isa(Groups.gens(A), Vector{Automorphism{1}})
+      @test length(Groups.gens(A)) == 1
       A = AutGroup(FreeGroup(1), special=true)
-      @test length(gens(A)) == 0
+      @test length(Groups.gens(A)) == 0
       A = AutGroup(FreeGroup(2))
-      @test length(gens(A)) == 7
-      gens = gens(A)
+      @test length(Groups.gens(A)) == 7
+      gens = Groups.gens(A)
 
       @test isa(A(Groups.rmul_autsymbol(1,2)), Automorphism)
       @test A(Groups.rmul_autsymbol(1,2)) in gens
@@ -146,15 +146,17 @@
       b = Groups.flip_autsymbol(2)*A(inv(Groups.rmul_autsymbol(1,2)))
       @test a*b == b*a
       @test a^3 * b^3 == A()
-      g,h = gens(A)[[1,8]] # (g, h) = (ϱ₁₂, ϱ₃₂)
+      g,h = Groups.gens(A)[[1,8]] # (g, h) = (ϱ₁₂, ϱ₃₂)
 
       @test Groups.domain(A) == NTuple{4, FreeGroupElem}(gens(A.objectGroup))
 
       @test (g*h)(Groups.domain(A)) == (h*g)(Groups.domain(A))
-      @test (g*h).savedhash != (h*g).savedhash
+      @test (g*h).savedhash == zero(UInt)
+      @test (h*g).savedhash == zero(UInt)
       a = g*h
       b = h*g
-      @test hash(a) == hash(b)
+      @test hash(a) != zero(UInt)
+      @test hash(b) == hash(a)
       @test a.savedhash == b.savedhash
       @test length(unique([a,b])) == 1
       @test length(unique([g*h, h*g])) == 1
@@ -228,6 +230,9 @@
       @test Groups.linear_repr(ϱ₁₂^-1) == M
       @test Groups.linear_repr(λ₁₂^-1) == M
 
+      @test Groups.linear_repr(ϱ₁₂*λ₁₂^-1) == eye(N)
+      @test Groups.linear_repr(λ₁₂^-1*ϱ₁₂) == eye(N)
+
       M = eye(N)
       M[2,2] = -1
       ε₂ = G(Groups.flip_autsymbol(2))
@@ -235,7 +240,8 @@
       @test Groups.linear_repr(ε₂) == M
       @test Groups.linear_repr(ε₂^2) == eye(N)
 
-      M = [0.0 0.0 1.0; 1.0 0.0 0.0; 0.0 1.0 0.0]
+      M = [0 1 0; 0 0 1; 1 0 0]
+
       σ = G(Groups.perm_autsymbol([2,3,1]))
       @test Groups.linear_repr(σ) == M
       @test Groups.linear_repr(σ^3) == eye(3)
diff --git a/test/DirectProd-tests.jl b/test/DirectProd-tests.jl
index 3fb124b..439ba65 100644
--- a/test/DirectProd-tests.jl
+++ b/test/DirectProd-tests.jl
@@ -1,32 +1,36 @@
 @testset "DirectProducts" begin
 
-   G = PermutationGroup(3)
-   g = G([2,3,1])
-   F, a = FiniteField(2,3,"a")
-
    @testset "Constructors" begin
-      @test isa(Groups.DirectProductGroup(G,2), AbstractArray.Group)
-      @test isa(G×G, AbstractAlgebra.Group)
-      @test isa(Groups.DirectProductGroup(G,2), Groups.DirectProductGroup{Generic.PermGroup{Int64}})
+      G = PermutationGroup(3)
+      g = G([2,3,1])
 
-      GG = Groups.DirectProductGroup(G,2)
+      @test Groups.DirectProductGroup(G,2) isa AbstractAlgebra.Group
+      @test G×G isa AbstractAlgebra.Group
+      @test Groups.DirectProductGroup(G,2) isa Groups.DirectProductGroup{Generic.PermGroup{Int64}}
 
-      @test GG == Groups.DirectProductGroup(G,2)
+      @test (G×G)×G == DirectProductGroup(G, 3)
+      @test (G×G)×G == (G×G)×G
 
-      @test Groups.DirectProductGroupElem([G(), G()]) == GG()
-      @test GG(G(), G()) == GG()
+      F = GF(13)
+      FF = F×F
+      @test FF×F == F×FF
 
-      @test isa(GG([g, g^2]), GroupElem)
-      @test isa(GG([g, g^2]), Groups.DirectProductGroupElem{Generic.perm{Int64}})
+      GG = DirectProductGroup(G,2)
+
+      @test Groups.DirectProductGroupElem([G(), G()]) == (G×G)()
+      @test GG(G(), G()) == (G×G)()
+
+      @test GG([g, g^2]) isa GroupElem
+      @test GG([g, g^2]) isa Groups.DirectProductGroupElem{Generic.perm{Int64}}
 
       h = GG([g,g^2])
 
       @test h == GG(h)
 
-      @test isa(GG(g, g^2), GroupElem)
-      @test isa(GG(g, g^2), Groups.DirectProductGroupElem)
+      @test GG(g, g^2) isa GroupElem
+      @test GG(g, g^2) isa Groups.DirectProductGroupElem
 
-      @test_throws String GG(g,g,g)
+      @test_throws DomainError GG(g,g,g)
       @test GG(g,g^2) == h
 
       @test size(h) == (2,)
@@ -34,46 +38,153 @@
       @test h[2] == g^2
       h[2] = G()
       @test h == GG(g, G())
+
    end
 
-   GG = Groups.DirectProductGroup(G,2)
-   FF = Groups.DirectProductGroup(F,2)
-
-   @testset "Types" begin
-      @test elem_type(GG) == Groups.DirectProductGroupElem{elem_type(G)}
-      @test elem_type(FF) == Groups.DirectProductGroupElem{elem_type(F)}
-      @test parent_type(typeof(GG(g,g^2))) == Groups.DirectProductGroup{typeof(G)}
-      @test parent_type(typeof(FF(a,a^2))) == Groups.DirectProductGroup{typeof(F)}
-
-      @test isa(FF([0,1]), GroupElem)
-      @test isa(FF([0,1]), Groups.DirectProductGroupElem)
-      @test isa(FF([0,1]), Groups.DirectProductGroupElem{elem_type(F)})
-      @test_throws MethodError FF(1,0)
-   end
-
-   @testset "Group arithmetic" begin
+   @testset "Basic arithmetic" begin
+      G = PermutationGroup(3)
       g = G([2,3,1])
-      h = GG([g,g^2])
+      h = (G×G)([g,g^2])
 
-      @test h^2 == GG(g^2,g)
-      @test h^6 == GG()
+      @test h^2 == (G×G)(g^2,g)
+      @test h^6 == (G×G)()
 
       @test h*h == h^2
 
-      @test h*inv(h) == GG()
+      @test h*inv(h) == (G×G)()
+   end
 
-      @test FF([0,a])*FF([a,1]) == FF(a,1+a)
+   @testset "elem/parent_types" begin
+      G = PermutationGroup(3)
+      g = G([2,3,1])
+
+      @test elem_type(G×G) == DirectProductGroupElem{elem_type(G)}
+      @test parent_type(typeof((G×G)(g,g^2))) == Groups.DirectProductGroup{typeof(G)}
+      @test parent((G×G)(g,g^2)) == DirectProductGroup(G,2)
+
+      F = GF(13)
+
+      @test elem_type(F×F) == DirectProductGroupElem{Groups.AddGrpElem{elem_type(F)}}
+      @test parent_type(typeof((F×F)(1,5))) == Groups.DirectProductGroup{AddGrp{typeof(F)}}
+      parent((F×F)(1,5)) == DirectProductGroup(F,2)
+   end
+
+   @testset "Additive/Multiplicative groups" begin
+
+      R, x = PolynomialRing(QQ, "x")
+      F, a = NumberField(x^3 + x + 1, "a")
+      G = PermutationGroup(3)
+
+      GG = Groups.DirectProductGroup(G,2)
+      FF = Groups.DirectProductGroup(F,2)
+
+      @testset "MltGrp basic functionality" begin
+         Gr = MltGrp(F)
+         @test Gr(a) isa MltGrpElem
+         g = Gr(a)
+         @test deepcopy(g) isa MltGrpElem
+         @test inv(g) == Gr(a^-1)
+         @test Gr() == Gr(1)
+         @test inv(g)*g == Gr()
+      end
+
+      @testset "AddGrp basic functionality" begin
+         Gr = AddGrp(F)
+         @test Gr(a) isa AddGrpElem
+         g = Gr(a)
+         @test deepcopy(g) isa AddGrpElem
+         @test inv(g) == Gr(-a)
+         @test Gr() == Gr(0)
+         @test inv(g)*g == Gr()
+      end
+   end
+
+   @testset "Direct Product of Multiplicative Groups" begin
+
+      R, x = PolynomialRing(QQ, "x")
+      F, a = NumberField(x^3 + x + 1, "a")
+      FF = Groups.DirectProductGroup(MltGrp(F),2)
+
+      @test FF([a,1]) isa GroupElem
+      @test FF([a,1]) isa DirectProductGroupElem
+      @test FF([a,1]) isa DirectProductGroupElem{MltGrpElem{elem_type(F)}}
+      @test_throws DomainError FF(1,0)
+      @test_throws DomainError FF([0,1])
+      @test_throws DomainError FF([1,0])
+
+      @test MltGrp(F) isa AbstractAlgebra.Group
+      @test MltGrp(F) isa MultiplicativeGroup
+      @test DirectProductGroup(MltGrp(F), 2) isa AbstractAlgebra.Group
+      @test DirectProductGroup(MltGrp(F), 2) isa DirectProductGroup{MltGrp{typeof(F)}}
+
+      F, a = NumberField(x^3 + x + 1, "a")
+      FF = DirectProductGroup(MltGrp(F), 2)
+
+      @test FF(a,a+1) == FF([a,a+1])
+      @test FF([1,a+1])*FF([a,a]) == FF(a,a^2+a)
       x, y = FF([1,a]), FF([a^2,1])
-      @test x*y == FF([a^2+1, a+1])
-      @test inv(x) == FF([1,a])
+      @test x*y == FF([a^2, a])
+      @test inv(x) == FF([1,-a^2-1])
+
+      @test parent(x) == FF
+   end
+
+   @testset "Direct Product of Additive Groups" begin
+
+      R, x = PolynomialRing(QQ, "x")
+      F, a = NumberField(x^3 + x + 1, "a")
+
+      # Additive Group
+      @test AddGrp(F) isa AbstractAlgebra.Group
+      @test AddGrp(F) isa AdditiveGroup
+      @test DirectProductGroup(AddGrp(F), 2) isa AbstractAlgebra.Group
+      @test DirectProductGroup(AddGrp(F), 2) isa DirectProductGroup{AddGrp{typeof(F)}}
+
+      FF = DirectProductGroup(AdditiveGroup(F), 2)
+
+      @test FF([0,a]) isa AbstractAlgebra.GroupElem
+      @test FF(F(0),a) isa DirectProductGroupElem
+      @test FF(0,0) isa DirectProductGroupElem{AddGrpElem{elem_type(F)}}
+
+      @test FF(F(1),a+1) == FF([1,a+1])
+
+      @test FF([F(1),a+1])*FF([a,a]) == FF(1+a,2a+1)
+
+      x, y = FF([1,a]), FF([a^2,1])
+      @test x*y == FF(a^2+1, a+1)
+      @test inv(x) == FF([F(-1),-a])
+
+      @test parent(x) == FF
    end
 
    @testset "Misc" begin
-      @test order(GG) == 36
+      F = GF(5)
+
+
+      FF = DirectProductGroup(F,2)
+      @test order(FF) == 25
+
+      elts = vec(collect(elements(FF)))
+      @test length(elts) == 25
+      @test all([g*inv(g) for g in elts] .== FF())
+      @test all(inv(g*h) == inv(h)*inv(g) for g in elts for h in elts)
+
+
+      FF = DirectProductGroup(MultiplicativeGroup(F), 3)
       @test order(FF) == 64
 
+      elts = vec(collect(elements(FF)))
+      @test length(elts) == 64
+      @test all([g*inv(g) for g in elts] .== FF())
+      @test all(inv(g*h) == inv(h)*inv(g) for g in elts for h in elts)
+
+
+      G = PermutationGroup(3)
+      GG = Groups.DirectProductGroup(G,2)
+      @test order(GG) == 36
+
       @test isa([elements(GG)...], Vector{Groups.DirectProductGroupElem{elem_type(G)}})
-      elts = [elements(GG)...]
+      elts = vec(collect(elements(GG)))
 
       @test length(elts) == 36
       @test all([g*inv(g) for g in elts] .== GG())
diff --git a/test/FreeGroup-tests.jl b/test/FreeGroup-tests.jl
index 007b6dc..1022ba8 100644
--- a/test/FreeGroup-tests.jl
+++ b/test/FreeGroup-tests.jl
@@ -41,7 +41,7 @@ end
 end
 
 @testset "FreeGroup" begin
-   @test isa(FreeGroup(["s", "t"]), Group)
+   @test isa(FreeGroup(["s", "t"]), AbstractAlgebra.Group)
    G = FreeGroup(["s", "t"])
 
    @testset "elements constructors" begin
diff --git a/test/WreathProd-tests.jl b/test/WreathProd-tests.jl
index bf3e1ba..a1443a6 100644
--- a/test/WreathProd-tests.jl
+++ b/test/WreathProd-tests.jl
@@ -1,18 +1,19 @@
 @testset "WreathProducts" begin
    S_3 = PermutationGroup(3)
-   F, a = FiniteField(2,3,"a")
+   R, x = PolynomialRing(QQ, "x")
+   F, a = NumberField(x^2 + 1, "a")
    b = S_3([2,3,1])
 
    @testset "Constructors" begin
       @test isa(Groups.WreathProduct(F, S_3), AbstractAlgebra.Group)
       @test isa(Groups.WreathProduct(F, S_3), Groups.WreathProduct)
-      @test isa(Groups.WreathProduct(F, S_3), Groups.WreathProduct{AbstractAlgebra.FqNmodFiniteField})
+      @test isa(Groups.WreathProduct(F, S_3), Groups.WreathProduct{AddGrp{Generic.ResField{Generic.Poly{Rational{BigInt}}}}, Int64})
 
       aa = Groups.DirectProductGroupElem([a^0 ,a, a^2])
 
       @test isa(Groups.WreathProductElem(aa, b), AbstractAlgebra.GroupElem)
       @test isa(Groups.WreathProductElem(aa, b), Groups.WreathProductElem)
-      @test isa(Groups.WreathProductElem(aa, b), Groups.WreathProductElem{typeof(a)})
+      @test isa(Groups.WreathProductElem(aa, b), Groups.WreathProductElem{AddGrpElem{Generic.ResF{Generic.Poly{Rational{BigInt}}}}, Int64})
 
       B3 = Groups.WreathProduct(F, S_3)
 
@@ -23,26 +24,15 @@
       @test B3(b) == Groups.WreathProductElem(B3.N(), b)
       @test B3(aa) == Groups.WreathProductElem(aa, S_3())
 
-      g = B3(aa, b)
+      @test B3([a^0 ,a, a^2], perm"(1,2,3)") isa WreathProductElem
 
-      @test g.p == b
-      @test g.n == aa
-      h = deepcopy(g)
-
-      @test hash(g) == hash(h)
-
-      g.n[1] = a
-
-      @test g.n[1] == a
-      @test g != h
-
-      @test hash(g) != hash(h)
+      @test B3([a^0 ,a, a^2], perm"(1,2,3)") == B3(aa, b)
    end
 
    @testset "Types" begin
       B3 = Groups.WreathProduct(F, S_3)
 
-      @test elem_type(B3) == Groups.WreathProductElem{elem_type(F), Int}
+      @test elem_type(B3) == Groups.WreathProductElem{AddGrpElem{elem_type(F)}, Int}
 
       @test parent_type(typeof(B3())) == Groups.WreathProduct{parent_type(typeof(B3.N.group())), Int}
 
@@ -50,30 +40,64 @@
       @test parent(B3()) == B3
    end
 
-   @testset "Group arithmetic" begin
+   @testset "Basic operations on WreathProductElem" begin
+      aa = Groups.DirectProductGroupElem([a^0 ,a, a^2])
       B3 = Groups.WreathProduct(F, S_3)
+      g = B3(aa, b)
 
-      x = B3(B3.N([1,0,0]), B3.P([2,3,1]))
-      y = B3(B3.N([0,1,1]), B3.P([2,1,3]))
+      @test g.p == b
+      @test g.n == DirectProductGroupElem(AddGrpElem.(aa.elts))
 
-      @test x*y == B3(B3.N([0,0,1]), B3.P([3,2,1]))
-      @test y*x == B3(B3.N([0,0,1]), B3.P([1,3,2]))
+      h = deepcopy(g)
+      @test h == g
+      @test !(g === h)
 
-      @test inv(x) == B3(B3.N([0,0,1]), B3.P([3,1,2]))
-      @test inv(y) == B3(B3.N([1,0,1]), B3.P([2,1,3]))
+      g.n[1] = parent(g.n[1])(a)
 
-      @test inv(x)*y == B3(B3.N([1,1,1]), B3.P([1,3,2]))
-      @test y*inv(x) == B3(B3.N([0,1,0]), B3.P([3,2,1]))
+      @test g.n[1] == parent(g.n[1])(a)
+      @test g != h
+
+      @test hash(g) != hash(h)
+
+      g.n[1] = a
+      @test g.n[1] == parent(g.n[1])(a)
+      @test g != h
+
+      @test hash(g) != hash(h)
+   end
+
+
+   @testset "Group arithmetic" begin
+      B4 = Groups.WreathProduct(GF(3), PermutationGroup(4))
+
+      x = B4([0,1,2,0], perm"(1,2,3)(4)")
+      @test inv(x) == B4([1,0,2,0], perm"(1,3,2)(4)")
+
+      y = B4([1,0,1,2], perm"(1,4)(2,3)")
+      @test inv(y) == B4([1,2,0,2], perm"(1,4)(2,3)")
+
+      @test x*y == B4([0,2,0,2], perm"(1,3,4)(2)")
+
+      @test y*x == B4([1,2,2,2], perm"(1,4,2)(3)")
+
+
+      @test inv(x)*y == B4([2,1,2,2], perm"(1,2,4)(3)")
+
+      @test y*inv(x) == B4([1,2,1,0], perm"(1,4,3)(2)")
 
    end
 
    @testset "Misc" begin
-      B3 = Groups.WreathProduct(FiniteField(2,1,"a")[1], S_3)
-      @test order(B3) == 48
+      B3 = Groups.WreathProduct(GF(3), S_3)
+      @test order(B3) == 3^3*6
 
-      Wr = WreathProduct(PermutationGroup(2),S_3)
+      B3 = Groups.WreathProduct(MultiplicativeGroup(GF(3)), S_3)
+      @test order(B3) == 2^3*6
+
+      Wr = WreathProduct(PermutationGroup(2),PermutationGroup(4))
 
       @test isa([elements(Wr)...], Vector{Groups.WreathProductElem{Generic.perm{Int}, Int}})
+      @test order(Wr) == 2^4*factorial(4)
 
       elts = [elements(Wr)...]