[FORMAT] 5 Covering Subgroups

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# 5 Covering Subgroups

Let \X be a collection of groups closed under taking homomorphic images. An \X-covering subgroup of a group G is a subgroup E satisfying

(C) E is in X , and EV = U whenever E is contained in U is contained in G with U/V in X.

It follows from the definition that an \X-covering subgroup E of G is also \X-covering in every subgroup U of G that contains E, and an easy argument shows that E is an \X-projector of every such U, i.e., E satisfies

(P) EK/K is an \X-maximal subgroup of U/K whenever K is normal in U.

Gaschütz showed that if F is a locally defined formation, then every finite solvable group has an F-covering subgroup. Indeed, locally defined formations are the only formations with this property. For such formations the F-projectors and F-covering subgroups of a solvable group coincide and form a single conjugacy class of subgroups. (See DH for details.)

- `CoveringSubgroup1(` G`,` F`) O`
- `CoveringSubgroup2(` G`,` F`) O`
- `CoveringSubgroupWrtFormation(` G`,` F`) O`

  If F is a locally defined integrated formation in GAP and if G is a finite solvable group, then the command `CoveringSubgroup1(` G`,` F`)` returns an F-covering subgroup of G. The function `CoveringSubgroup2` uses a different algorithm to compute F-covering subgroups. The user may choose either function. Experiments with large groups suggest that `CoveringSubgroup1` is somewhat faster. `CoveringSubgroupWrtFormation` checks first to see if either of these two functions has already computed an F-covering subgroup of G and, if not, it calls `FCoveringGroup1` to compute one.

  Nilpotent-covering subgroups are also called Carter subgroups.

- `CarterSubgroup(` G`) A`

  The command `CarterSubgroup(` G`)` is equivalent to `CoveringSubgroupWrtFormation(` G`, Formation( "Nilpotent" ) )`.

  All of these functions call upon F-normalizer algorithms as subroutines.

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