indexing(n) = [(i,j) for i in 1:n for j in 1:n if i≠j]

function generating_set(G::AutGroup{N}, n=N) where N

    rmuls = [Groups.rmul_autsymbol(i,j) for (i,j) in indexing(n)]
    lmuls = [Groups.lmul_autsymbol(i,j) for (i,j) in indexing(n)]
    gen_set = G.([rmuls; lmuls])

    return [gen_set; inv.(gen_set)]
end

function Sq(RG::GroupRing{AutGroup{N}}) where N
    S₂ = generating_set(RG.group, 2)
    ℤ = Int64
    Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);

    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
    elt = sum(σ(Δ₂)^2 for σ in Alt_N)
    # return RG(elt.coeffs÷factorial(N-2))
    return elt
end

function Adj(RG::GroupRing{AutGroup{N}}) where N
    S₂ = small_generating_set(RG, 2)
    ℤ = Int64
    Δ₂ = length(T2)*one(RG, ℤ) - RG(S₂, ℤ);

    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]

    adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
    adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 1]

    @time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
    # return RG(elt.coeffs÷factorial(N-2)^2)
    return elt
end

function Op(RG::GroupRing{AutGroup{N}}) where N
    S₂ = small_generating_set(RG, 2)
    ℤ = Int64
    Δ₂ = length(T2)*one(RG, ℤ) - RG(S₂, ℤ);

    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]

    adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
    op(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 0]

    @time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in op(σ)) for σ in Alt_N);
    # return RG(elt.coeffs÷factorial(N-2)^2)
    return elt
end