GroupsWithPropertyT/groups/speciallinear.jl

struct SpecialLinearGroup{N} <: SymmetrizedGroup
    group::AbstractAlgebra.Group
    p::Int
    X::Bool
end

function SpecialLinearGroup(args::Dict)
    N = args["SL"]
    p = args["p"]
    X = args["X"]

    if p == 0
        G = MatrixSpace(Nemo.ZZ, N, N)
    else
        R = Nemo.NmodRing(UInt(p))
        G = MatrixSpace(R, N, N)
    end
    return SpecialLinearGroup{N}(G, p, X)
end

function name(G::SpecialLinearGroup{N}) where N
    if G.p == 0
       R = (G.X ? "Z[x]" : "Z")
    else
       R = "F$(G.p)"
    end
    return SL($(G.N),$R)
end

group(G::SpecialLinearGroup) = G.group

function generatingset(G::SpecialLinearGroup{N}) where N
    G.p > 0 && G.X && throw("SL(n, F_p[x]) not implemented")
    SL = group(G)
    return generatingset(SL, G.X)
end

# r is the injectivity radius of
# SL(n, Z[X]) -> SL(n, Z) induced by X -> 100

function generatingset(SL::MatSpace, X::Bool=false, r=5)
    n = SL.cols
    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]

    if !X
        S = [E(idx[1],idx[2],SL) for idx in indexing]
    else
        S = [E(i,j,SL,v) for (i,j) in indexing for v in [1, 100*r]]
    end
    return unique([S; inv.(S)])
end

function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
    @assert i≠j
    m = one(M)
    m[i,j] = val
    return m
end

function autS(G::SpecialLinearGroup{N}) where N
    return WreathProduct(PermutationGroup(2), PermutationGroup(N))
end