Let `X` be a collection of groups closed under taking homomorphic images.
An ** X-covering subgroup** of a group

(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

(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( `

`, `

` ) O`

`CoveringSubgroup2( `

`, `

` ) O`

`CoveringSubgroupWrtFormation( `

`, `

` ) 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( `

` ) 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 2011