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( `

`, `

` ) 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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FORMAT manual

March 2018