| products/sources/formale Sprachen/GAP/pkg/crisp/doc/ (Browser von der Mozilla Stiftung Version 136.0.1©) Datei vom 20.1.2016 mit Größe 4 kB |
|
Quelle examples.tex Sprache: Latech |
|||||||||
|
%!TEX root = manual.tex
% !TeX TS-program = pdftex % !TEX spellcheck =en_GB %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% examples.tex CRISP documentation Burkhard Höfling %% %% Copyright © 2000, Burkhard Höfling %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 $p$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E %% ¤ Dauer der Verarbeitung: 0.23 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
|
2026-03-28