Normal Structure and Characteristic Subgroups

Subsections

• Characteristic Subgroups and Subgroup Series
• Subgroup Structure

Characteristic Subgroups and Subgroup Series

Agemo(G, i) : GrpAb, RngIntElt -> GrpAb

Given a finite 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.

Centre(G) : GrpAb -> GrpAb

Center(G) : GrpAb -> GrpAb

The centre of the group G.

ChiefSeries(G) : GrpAb -> [GrpAb]

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

DerivedLength(G) : GrpAb -> RngIntElt

The derived length of the group G.

DerivedSeries(G) : GrpAb -> [GrpAb]

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

DerivedSubgroup(G) : GrpAb -> GrpAb

DerivedGroup(G) : GrpAb -> GrpAb

The derived subgroup of the group G.

ElementaryAbelianSeries(G) : GrpAb -> [GrpAb]

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.

FittingSubgroup(G) : GrpAb -> GrpAb

The Fitting subgroup of the group G.

FrattiniSubgroup(G) : GrpAb -> GrpAb

The Frattini subgroup of the group G.

Hypercentre(G) : GrpAb -> GrpAb

Hypercenter(G) : GrpAb -> GrpAb

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

NilpotencyClass(G) : GrpAb -> RngIntElt

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

Omega(G, i) : GrpAb, RngIntElt -> GrpAb

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

SubnormalSeries(G, H) : GrpAb, GrpAb -> [GrpAb]

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.

UpperCentralSeries(G) : GrpAb -> [GrpAb]

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

Subgroup Structure

MaximalSubgroups(G) : GrpAb -> [GrpAb]

The maximal subgroups of the finite group G returned as a sequence of subgroups.

Subgroups(G:parameters) : GrpAb -> [Rec]

The subgroups of the finite group G returned as a sequence of records. The record fields are subgroup, storing the actual group; order, storing the group order; and length, storing the length of the conjugacy class, which is always 1 for abelian groups.

     Sub: [RngIntElt]                    Default: []

If parameter Sub is set, only subgroups with invariants equal to the given sequence are found. The given sequence should contain positive integers, such that each divides the following.

     Quot: [RngIntElt]                   Default: []

If parameter Quot is set, only subgroups such that the quotient group has invariants equal to the given sequence are found. The given sequence should contain positive integers, such that each divides the following.

Example GrpAb_Subgroups (H25E4)

> G := AbelianGroup([2,6]);
> s := Subgroups(G); #s;
10
> s[7];
rec<recformat<order, length, subgroup, 
presentation> | order := 3, length := 1, subgroup 
:= Abelian Group isomorphic to Z/3
Defined on 1 generator in supergroup G:
  .1 = 2 * G.2
Relations:
  3 *.1 = 0>
> [x`order:x in s];
[ 12, 6, 4, 2, 6, 6, 3, 2, 2, 1 ]

> s22 := Subgroups(G:Sub := [2,2]); #s22;
1
> s22;
Conjugacy classes of subgroups
------------------------------

[1]     Order 4            Length 1
        Abelian Group isomorphic to Z/2 + Z/2
        Defined on 2 generators in supergroup G:
          .1 = G.1
          .2 = 3*G.2
        Relations:
          2*.1 = 0
          2 *.2 = 0

> q2 := Subgroups(G:Quot := [2]); #q2;
3
> q2[3]`subgroup;
Abelian Group isomorphic to Z/6
Defined on 1 generator in supergroup G:
  .1 = G.1 + G.2
Relations:
  6 *.1 = 0