Subsections

• Characteristic Subgroups and Subgroup Series
• The Abstract Structure of a Group

Center(G) : GrpMat -> GrpMat

Construct the centre of the group G.

The derived length of G. If G is non-soluble, the function returns the number of terms in the series terminating with the soluble residual.

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

DerivedGroup(G) : GrpMat -> GrpMat

The derived subgroup of the group G.

The Fitting subgroup of the group G.

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

The nilpotency class of the group G.

NormalClosure(G, H) : GrpMat -> GrpMat

The normal closure of the subgroup H of group G.

The normal subgroup lattice of G. The subgroups are found by first determining the minimal normals using the conjugacy classes of G and then extending these layer by layer until G is reached. The algorithm is thus only practical in the case of small to medium sized groups.

SolvableResidual(G) : GrpMat -> GrpMat

The solvable residual of the group G.

Given a group G and a subnormal 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. As the algorithm used requires the conjugacy classes of G, this function is much more restricted in its range of application than DerivedSeries and LowerCentralSeries.

> G := MatrixGroup< 3, GF(5) | [0,1,0, 1,0,0, 0,0,1], 
>                                      [0,1,0, 0,0,1, 1,0,0],
>                                      [2,0,0, 0,1,0, 0,0,1] >;
> Order(G);
384
> DerivedGroup(G);
MatrixGroup(3, GF(5, 1))
Generators:
[0 0 1]
[1 0 0]
[0 1 0]


[2 0 0]
[0 3 0]
[0 0 1]
> D := DerivedSeries(G);
> [ Order(d) : d in D ];
[ 384, 48, 16, 1 ]
> L := LowerCentralSeries(G);
> [ Order(l) : l in L ];
[ 384, 48 ]
> K := sub< G | [ 2,0,0, 0,3,0, 0,0,2 ] >;
> S := SubnormalSeries(G, K);
> [ Order(s) : s in S ];
[ 384, 16, 4 ]

Invariants(G) : GrpMat -> [ RngIntElt ]

Given an abelian group G, return a sequence Q containing the types of each p-primary component of G.

Given a matrix group G, return a sequence S of tuples that represent the composition factors of G, ordered according to some composition series of G. Each tuple is a triple of integers f, d, q that defines the isomorphism type of the corresponding composition factor. A triple < f, d, q > describes a simple group as follows. The integer f defines the family to which the group belongs, and d and q are the parameters of the family. The length of the sequence S is the number of composition factors of G. The numbering of the simple group families is given in Tables 1 and 2 of the chapter on permutation groups.