The functions given here all assume that G is a soluble group having order p_(1)^(e_1)p_(2)^(e_2) ... p_(k)^(e_k).

ComplementBasis(G) : GrpPC -> [GrpPC]

A complement basis of the soluble group G. This is a sequence of k subgroups of G, where the i-th subgroup has order p_(1)^(e_(1)) ... p_(i - 1)^ (e_(i - 1)) p_(i + 1)^(e_(i + 1)) ... p_(k)^(e_(k)), i.e. the complements of the k Sylow subgroups of G.

HallSubgroup(G, S) : GrpPC, RngIntElt -> GrpPC

The Hall pi-subgroup of G, where pi is defined by S. The argument S may be a set of integers taken from the set {p_(1), ..., p_(k)}, or a single integer taken from the set {p_(1), ..., p_(k), - p_(1), ..., - p_(k)}. If S = - p_(i), then the Hall p_(i)'-subgroup of G is returned.

pCore(G, S) : GrpPC, RngIntElt -> GrpPC

The core of the Hall pi-subgroup, where pi is defined by the argument S, which has the same interpretation as for HallSubgroup above.

A Sylow basis for the soluble group G. This is a sequence of k subgroups of G, having orders p_(1)^(e_(1)), ..., p_(k)^(e_(k)), i.e. the k Sylow subgroups of G.

Sylow(G, p) : GrpPC, RngIntElt -> GrpPC

A Sylow p-subgroup for the group G.

SystemNormaliser(G) : GrpPC -> GrpPC

The system normalizer for the group G. The system normalizer of the complement basis Sigma = { H_1, ..., H_k } is defined to be the intersection of the normalizers in G of each H_i, ie. N(Sigma) = cap_(i=1)^(k) N_G(H_i). The algorithm used is derived directly from the definition.

> H := DihedralGroup(GrpPerm, 5);
> G := WreathProduct(DihedralGroup(GrpPC, 3), DihedralGroup(GrpPC, 5),
>         [H.2, H.1]);
> H2 := HallSubgroup(G, 2);

> H35 := HallSubgroup(G, -2);