Subsections

• Characteristic Subgroups and Subgroup Series
• Subgroup Structure

Given a p-group G, return the characteristic subgroup of G generated by the elements x^(p^i), x in G, where i is a positive integer.

Given an abelian group G, return sequence B and I such that ( order)(B[i]) = I[i] and < B > = G and I are the types of each p-primary component of G.

The abelian invariants of the abelian group G as a sequence of integers.

Center(G) : GrpPC -> GrpPC

The centre of the group G.

A chief series for the group G. The series is returned as a sequence of subgroups of G.

Given a normal subgroup N of a group G, return conjugacy class representatives for the complements of N in G.

A composition series for the group G. The series is returned as a sequence of subgroups of G. The i-th term of the composition series has a presentation given by the generators G.i through G.NPCgens(G) and relations involving those generators only.

CompositionSeries(G, i) : GrpPC, RngIntElt -> [GrpPC]

The i + 1-th entry of the composition series for the group G. Its presentation given by the generators G.i + 1 through G.NPCgens(G) and relations involving those generators only.

The derived length of the group G.

The derived series of the group G. The series is returned as a sequence of subgroups.

DerivedGroup(G) : GrpPC -> GrpPC

The derived subgroup of the group G.

An elementary abelian series is a chain of normal subgroups with the property that the quotient of each pair of successive terms in the series is elementary abelian. Thus, it refines the derived series. The elementary abelian series for the group G is returned as a sequence of subgroups.

The Fitting subgroup of the group G.

The Frattini subgroup of the group G.

Hypercenter(G) : GrpPC -> GrpPC

The hypercentre of the group G, i.e. the stationary term in the upper central series for G.

Given a p-group G, return the Jennings series for G. The series is returned as a sequence of subgroups. The Jennings series J_1 triangleright J_2 triangleright ... triangleright J_i ... of a p-group G is defined inductively as follows:

• J_1 = G,
• J_(i + 1) = <(J_i, G), J_k^p>, with k = Ceiling((i + 1)/p), i > 0.

The lower central series for the group G. The series is returned as a sequence of subgroups.

The nilpotency class of the group G. If G is not nilpotent, then -1 is returned.

Given a p-group G, return the characteristic subgroup of G generated by the elements of order dividing p^i, where i is a positive integer.

The p-central series for G, where p is a prime dividing |G|. The series is returned as a sequence of subgroups. The p-central series P_1 triangleright P_2 triangleright ... triangleright P_i of a soluble group G is defined inductively as follows:

• P_1 = G,
• P_(i + 1) = (G, P_i)P_i^p, for i > 0.

Given a group G and a subgroup H of G, return a sequence of subgroups commencing with G and terminating with H, such that each subgroup is normal in the previous one. If H is not subnormal in G, the empty sequence is returned.

The upper central series of G. The series is returned as a sequence of subgroups.

> H := DihedralGroup(GrpPerm, 5);
> G := WreathProduct(DihedralGroup(GrpPC, 3), DihedralGroup(GrpPC, 5),
>       [H.2, H.1]);
> EAS := ElementaryAbelianSeries(G);
> for i := 1 to #EAS do
>    print FactoredOrder(EAS[i]);
> end for;
	[ <2, 6>, <3, 5>, <5, 1> ]
	[ <2, 4>, <3, 5>, <5, 1> ]
	[ <2, 4>, <3, 5> ]
	[ <3, 5> ]
	[]

An elementary abelian minimal normal subgroup of the soluble group G.

The maximal subgroups of G returned as a sequence.