%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% examples.tex CRISP documentation Burkhard H\"ofling %% %% @(#)$Id: examples.tex,v 1.5 2002/01/16 12:23:06 gap Exp $ %% %% Copyright (C) 2000, Burkhard H\"ofling, Mathematisches Institut, %% Friedrich Schiller-Universit\"at Jena, Germany %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Examples of group classes} This chapter describes some pre-defined group classes, namely the classes of all abelian, nilpotent, and supersolvable 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Pre-defined group classes}\null \>`TrivialGroups'{class}!{of all trivial groups} V \index{trivial groups!class of}% \index{class!of all trivial groups} 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 \index{nilpotent groups!class of}% \index{class!of all nilpotent groups}% This global variable contains the class of all finite nilpotent groups. It is a subgroup closed saturated Fitting formation. \>`SupersolvableGroups'{class}!{of all supersolvable groups} V \index{supersolvable groups!class of}% \index{class!of all supersolvable groups}% This global variable contains the class of all finite supersolvable groups. It is a subgroup closed saturated formation. \>`AbelianGroups'{class}!{of all abelian groups} V \indextt{AbelianGroups} \index{abelian groups!class of}% \index{class!of all abelian groups}% is the class of all abelian groups. It is a subgroup closed formation. \>AbelianGroupsOfExponent(<n>) F \index{class!of all abelian groups of bounded exponent} \index{abelian groups of bounded exponent!class of} 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} 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} returns the class of all <p>-groups, where <p> is a prime. The result is a subgroup-closed saturated Fitting formation. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Pre-defined projector functions}\null \>NilpotentProjector(<grp>) A \index{Carter subgroup}% 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 solvable group equal its a Carter subgroups, that is, its self-normalizing nilpotent subgroups. \>SupersolvableProjector(<grp>) A These functions return a projector for the class of all finite supersolvable groups. For a definition, see "Projector". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Pre-defined sets of primes}\null \>`AllPrimes'{set}!{of all primes} V \index{primes!set of all}% \label{AllPrimes}% 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 $p$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E %%