# 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 in X, and EV = U whenever E le U le 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, then 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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November 2000