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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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