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# 6 Examples of group classes

### Sections

- Pre-defined group classes
- Pre-defined projector functions
- Pre-defined sets of primes

This chapter describes some pre-defined
group classes, namely the classes of all abelian, nilpotent, and supersoluble
groups. Moreover, there are some functions constructing the classes of all
*p*-groups, π-groups, and abelian groups whose exponent divides a given
positive integer.

The definitions of these group classes can also serve as further examples of
how group classes can be defined using the methods described in the preceding
chapters.

`TrivialGroups V`

The global variable `TrivialGroups`

contains the class of all trivial groups. It is a
subgroup closed saturated Fitting formation.

`NilpotentGroups V`

This global variable contains the class of all finite nilpotent groups. It is a
subgroup closed saturated Fitting formation.

`SupersolubleGroups V`

`SupersolvableGroups V`

This global variable contains the class of all finite supersoluble groups. It
is a subgroup closed saturated formation.

`AbelianGroups V`

is the class of all abelian groups. It is a subgroup closed formation.

`AbelianGroupsOfExponent(`

`n``) F`

returns the class of all abelian groups of exponent dividing `n`,
where `n` is
a positive integer. It is always a subgroup-closed formation.

`PiGroups(`

`pi``) F`

constructs the class of all `pi`-groups. `pi` may be a non-empty class or a
set of primes. The result is a subgroup-closed saturated Fitting formation.

`PGroups(`

`p``) F`

returns the class of all `p`-groups, where `p` is a prime. The result is a
subgroup-closed saturated Fitting formation.

`NilpotentProjector(`

`grp``) A`

This function returns a projector for the class of all finite nilpotent
groups. For a definition, see Projector. Note that the nilpotent projectors
of a finite soluble group equal its a Carter subgroups, that is, its
self-normalizing nilpotent subgroups.

`SupersolubleProjector(`

`grp``) A`

`SupersolvableProjector(`

`grp``) A`

These functions return a projector for the class of all finite supersoluble
groups. For a definition, see Projector.

`AllPrimes V`

labelAllPrimesrelax
is the set of all (integral) primes. This should be
installed as value for `Characteristic(`

`grpclass``)`

if the group class
`grpclass` contains cyclic groups of prime order *p* for arbitrary primes *p*.

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CRISP manual

March 2016