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\Chapter{Examples of group classes}

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, $\pi$-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.

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\Section{Pre-defined group classes}\null

\>`TrivialGroups'{class}!{of all trivial groups} V

\indextt{TrivialGroups}\relax
\index{trivial groups!class of}\relax
\index{class!of all trivial groups}\relax
The global variable `TrivialGroups' contains the class of all trivial groups. It is a
subgroup closed saturated Fitting formation.

\>`NilpotentGroups'{class}!{of all nilpotent groups} V

\indextt{NilpotentGroups}\relax
\index{nilpotent groups!class of}\relax
\index{class!of all nilpotent groups}\relax
This global variable contains the class of all finite nilpotent groups. It is a
subgroup closed saturated Fitting formation.


\>`SupersolubleGroups'{class}!{of all supersoluble groups} V
\>`SupersolvableGroups'{class}!{of all supersoluble groups} V

\indextt{SupersolubleGroups}\relax
\indextt{SupersolvableGroups}\relax
\index{supersoluble groups!class of}\relax
\index{class!of all supersoluble groups}\relax
This global variable contains the class of all finite supersoluble groups. It
is a subgroup closed saturated formation.


\>`AbelianGroups'{class}!{of all abelian groups} V

\indextt{AbelianGroups}\relax
\index{abelian groups!class of}\relax
\index{class!of all abelian groups}\relax
is the class of all abelian groups. It is a subgroup closed formation.


\>AbelianGroupsOfExponent(<n>) F

\indextt{AbelianGroupsOfExponent}\relax
\index{class!of all abelian groups of bounded exponent}\relax
\index{abelian groups of bounded exponent!class of}\relax
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

\index{class!of all $\pi$-groups}\relax
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

\index{class!of all $p$-groups}\relax
returns the class of all <p>-groups, where <p> is a prime.  The result is a
subgroup-closed saturated Fitting formation.

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\Section{Pre-defined projector functions}\null

\>NilpotentProjector(<grp>) A

\index{Carter subgroup}\relax
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". 

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\Section{Pre-defined sets of primes}\null

\>`AllPrimes'{set}!{of all primes} V

\indextt{AllPrimes}\relax
\index{primes!set of all}\relax
\label{AllPrimes}\relax
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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