@testset "Automorphisms" begin

    @testset "Transvections" begin

        @test Groups.Transvection(:ϱ, 1, 2) isa Groups.GSymbol
        @test Groups.Transvection(:ϱ, 1, 2) isa Groups.Transvection
        @test Groups.Transvection(:λ, 1, 2) isa Groups.GSymbol
        @test Groups.Transvection(:λ, 1, 2) isa Groups.Transvection
        t = Groups.Transvection(:ϱ, 1, 2)
        @test inv(t) isa Groups.GSymbol
        @test inv(t) isa Groups.Transvection

        @test t != inv(t)

        s = Groups.Transvection(:ϱ, 1, 2)
        @test t == s
        @test hash(t) == hash(s)

        s_ = Groups.Transvection(:ϱ, 1, 3)
        @test s_ != s
        @test hash(s_) != hash(s)

        @test Groups.gersten_alphabet(3) isa Alphabet
        A = Groups.gersten_alphabet(3)
        @test length(A) == 12

        @test sprint(show, Groups.ϱ(1, 2)) == "ϱ₁.₂"
        @test sprint(show, Groups.λ(3, 2)) == "λ₃.₂"
    end

    A4 = Alphabet(
        [:a, :A, :b, :B, :c, :C, :d, :D],
        [2, 1, 4, 3, 6, 5, 8, 7]
    )

    A5 = Alphabet(
        [:a, :A, :b, :B, :c, :C, :d, :D, :e, :E],
        [2, 1, 4, 3, 6, 5, 8, 7, 10, 9]
    )

    F4 = FreeGroup(A4)
    a, b, c, d = gens(F4)
    D = ntuple(i -> gens(F4, i), 4)

    @testset "Transvection action correctness" begin
        i, j = 1, 2
        r = Groups.Transvection(:ϱ, i, j)
        l = Groups.Transvection(:λ, i, j)

        (t::Groups.Transvection)(v::Tuple) = Groups.evaluate!(v, t)

        @test r(deepcopy(D)) == (a * b, b, c, d)
        @test inv(r)(deepcopy(D)) == (a * b^-1, b, c, d)
        @test l(deepcopy(D)) == (b * a, b, c, d)
        @test inv(l)(deepcopy(D)) == (b^-1 * a, b, c, d)

        i, j = 3, 1
        r = Groups.Transvection(:ϱ, i, j)
        l = Groups.Transvection(:λ, i, j)
        @test r(deepcopy(D)) == (a, b, c * a, d)
        @test inv(r)(deepcopy(D)) == (a, b, c * a^-1, d)
        @test l(deepcopy(D)) == (a, b, a * c, d)
        @test inv(l)(deepcopy(D)) == (a, b, a^-1 * c, d)

        i, j = 4, 3
        r = Groups.Transvection(:ϱ, i, j)
        l = Groups.Transvection(:λ, i, j)
        @test r(deepcopy(D)) == (a, b, c, d * c)
        @test inv(r)(deepcopy(D)) == (a, b, c, d * c^-1)
        @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.Transvection(:ϱ, i, j)
        l = Groups.Transvection(:λ, i, j)
        @test r(deepcopy(D)) == (a, b * d, c, d)
        @test inv(r)(deepcopy(D)) == (a, b * d^-1, c, d)
        @test l(deepcopy(D)) == (a, d * b, c, d)
        @test inv(l)(deepcopy(D)) == (a, d^-1 * b, c, d)
    end

    A = SpecialAutomorphismGroup(F4, max_rules=1000)

    @testset "AutomorphismGroup constructors" begin
        @test A isa Groups.AbstractFPGroup
        @test A isa AutomorphismGroup
        @test alphabet(A) isa Alphabet
        @test Groups.relations(A) isa Vector{<:Pair}
        @test sprint(show, A) == "automorphism group of free group on 4 generators"
    end

    @testset "Automorphisms: hash and evaluate" begin
        @test Groups.domain(gens(A, 1)) == D
        g, h = gens(A, 1), gens(A, 8) # (ϱ₁.₂, ϱ₃.₂)

        @test evaluate(g * h) == evaluate(h * g)
        @test (g * h).savedhash == zero(UInt)

        @test contains(sprint(show, typeof(g)), "Automorphism{FreeGroup{Symbol")

        a = g * h
        b = h * g
        @test hash(a) != zero(UInt)
        @test hash(a) == hash(b)
        @test a.savedhash == b.savedhash

        @test length(unique([a, b])) == 1
        @test length(unique([g * h, h * g])) == 1

        # Not so simple arithmetic: applying starting on the left:
        # ϱ₁₂*ϱ₂₁⁻¹*λ₁₂*ε₂ == σ₂₁₃₄

        g = gens(A, 1)
        x1, x2, x3, x4 = Groups.domain(g)
        @test evaluate(g) == (x1 * x2, x2, x3, x4)

        g = g * inv(gens(A, 4)) # ϱ₂₁
        @test evaluate(g) == (x1 * x2, x1^-1, x3, x4)

        g = g * gens(A, 13)
        @test evaluate(g) == (x2, x1^-1, x3, x4)
    end

    @testset "Automorphisms: SAut(F₄)" begin
        N = 4
        G = SpecialAutomorphismGroup(FreeGroup(N))

        S = gens(G)
        @test S isa Vector{<:FPGroupElement{<:AutomorphismGroup{<:FreeGroup}}}

        @test length(S) == 2 * N * (N - 1)
        @test length(unique(S)) == length(S)

        S_sym = [S; inv.(S)]
        @test length(S_sym) == length(unique(S_sym))

        pushfirst!(S_sym, one(G))

        B_2 = [i * j for (i, j) in Base.product(S_sym, S_sym)]
        @test length(B_2) == 2401
        @test length(unique(B_2)) == 1777

        @test all(g -> isone(inv(g) * g), B_2)
        @test all(g -> isone(g * inv(g)), B_2)
    end

    @testset "Forward evaluate" begin
        N = 3
        F = FreeGroup(N)
        G = SpecialAutomorphismGroup(F)

        a = gens(G, 1) # ϱ₁₂

        f = gens(F)

        @test a(f[1]) == f[1] * f[2]
        @test all(a(f[i]) == f[i] for i in 2:length(f))

        S = let s = gens(G)
            [s; inv.(s)]
        end

        @test all(
            map(first(Groups.wlmetric_ball(S, radius=2))) do g
                lm = Groups.LettersMap(g)
                img = evaluate(g)

                fimg = [F(lm[first(word(s))]) for s in gens(F)]

                succeeded = all(img .== fimg)
                @assert succeeded "forward evaluation of $(word(g)) failed: \n img=$img\n fimg=$(tuple(fimg...))"
                succeeded
            end
        )
    end

    Logging.with_logger(Logging.NullLogger()) do
        @testset "GroupsCore conformance" begin
            test_Group_interface(A)
            g = A(rand(1:length(alphabet(A)), 10))
            h = A(rand(1:length(alphabet(A)), 10))

            test_GroupElement_interface(g, h)
        end
    end
end