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# 6 Examples of group classes

### Sections

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.

## 6.1 Pre-defined group classes

• `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.

## 6.2 Pre-defined projector functions

• `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.

## 6.3 Pre-defined sets of primes

• `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