[CRISP] 7 Lists of normal subgroups

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# 7 Lists of normal subgroups

### Sections

1. Functions for normal and characteristic subgroups
2. Functions for the socle of finite groups

The algorithms in CRISP can also be used to compute certain normal subgroups of a finite soluble group efficiently. In particular, CRISP provides fast methods for computing all normal subgroups, all minimal normal subgroups, and the socle of a finite soluble group.

## 7.1 Functions for normal and characteristic subgroups

- `NormalSubgroups(`grp`) A`

  For finite soluble groups grp, CRISP provides an efficient method to compute `NormalSubgroups` (see NormalSubgroups).

- `CharacteristicSubgroups(`grp`) A`

  returns a list containing all characteristic subgroups of the finite soluble group grp. `CharacteristicSubgroups` calls `AllInvSgrsWithQPropUnderAction`.

- `MinimalNormalSubgroups(`grp`) A`

  CRISP provides an efficient method to compute a list of all minimal normal subgroups of grp (see MinimalNormalSubgroups).

- `MinimalNormalPSubgroups(`grp`,` p`) A`

  For a prime p, this function computes a list of all p-subgroups which are minimal among the nontrivial normal subgroups of grp.

- `AbelianMinimalNormalSubgroups(`grp`) A`

  This computes a list of all minimal normal subgroups of grp which are abelian. If grp is soluble, this list coincides with the list of all minimal normal subgroups of grp.

  ## 7.2 Functions for the socle of finite groups

- `Socle(`grp`) A`

  CRISP provides a method for `Socle` (see Socle) for which works for all finite soluble groups grp. The socle of a group grp is the subgroup generated by all minimal normal subgroups of grp. See also SolubleSocle and PSocle below.

      gap> Size(Socle( DirectProduct(DihedralGroup(8), CyclicGroup(12))));
      12

- `AbelianSocle(`grp`) A`
- `SolubleSocle(`grp`) A`
- `SolvableSocle(`grp`) A`

  This function computes the soluble socle of grp. The soluble socle of a group grp is the subgroup generated by all minimal normal soluble subgroups of grp.

- `SocleComponents(`grp`) A`

  This function returns a list of minimal normal subgroups of grp such that the socle of grp (see Socle) is the direct product of these minimal normal subgroups. Note that, in general, this decomposition is not unique. Currently, this function is only implemented for finite soluble groups. See also SolubleSocleComponents and PSocleComponents.

- `AbelianSocleComponents(`grp`) A`
- `SolubleSocleComponents(`grp`) A`
- `SolvableSocleComponents(`grp`) A`

  This function returns a list of soluble minimal normal subgroups of grp such that the socle of grp (see Socle) is the direct product of these minimal normal subgroups. Note that, in general, this decomposition is not unique.

- `PSocle(`grp`,` p`) A`

  If p is a prime, the p-socle of a group grp is the subgroup generated by all minimal normal p-subgroups of grp.

- `PSocleComponents(`grp`,` p`) A`

  For a prime p, this function returns a list of minimal normal p-subgroups of grp such that the p-socle of grp (see PSocle) is the direct product of these minimal normal subgroups. Note that, in general, this decomposition is not unique.

- `PSocleSeries(`grp`,` p`) A`

  For a prime p, this function returns an ascending grp-composition series of the p-socle of grp.

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