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% !TeX TS-program = pdftex % !TEX spellcheck =en_GB %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% normal.tex CRISP documentation Burkhard Höfling %% %% Copyright © 2015 Burkhard Höfling %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Lists of normal subgroups} The algorithms in {\CRISP} can also be used to compute certain normal subgroups of a finite solu group efficiently. In particular, {\CRISP} provides fast methods for computing all normal su %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Functions for normal and characteristic subgroups}\null \>NormalSubgroups(<grp>) A For finite soluble groups <grp>, {\CRISP} provides an efficient method to compute `NormalSubgr \>CharacteristicSubgroups(<grp>) A returns a list containing all characteristic subgroups of the finite soluble group~<grp>. `CharacteristicSubgroups' calls `AllInvSgrsWithQPropUnderAction'. \>MinimalNormalSubgroups(<grp>) A \index{minimal normal subgroups}\relax {\CRISP} provides an efficient method to compute a list of all minimal normal subgroups of <grp> ( \>MinimalNormalPSubgroups(<grp>, <p>) A \index{minimal normal $p$-subgroups}\relax For a prime <p>, this function computes a list of all <p>-subgroups which are minimal among the nontr normal subgroups of <grp>. \>AbelianMinimalNormalSubgroups(<grp>) A \index{minimal normal subgroups}\relax This computes a list of all minimal normal subgroups of <grp> which are abelian. If <grp> is soluble, minimal normal subgroups of <grp>. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Functions for the socle of finite groups}\null \>Socle(<grp>) A {\CRISP} provides a method for `Socle' (see "ref:Socle") for which works for all finite soluble groups <grp>. The socle of a group <grp> is the subgroup generated by all minimal normal subgroups of <grp>. See also "SolubleSocle" and "PSocle" below. \beginexample gap> Size(Socle( DirectProduct(DihedralGroup(8), CyclicGroup(12)))); 12 \endexample \>AbelianSocle(<grp>) A \>SolubleSocle(<grp>) A \>SolvableSocle(<grp>) A This function computes the soluble socle of <grp>. The soluble socle of a group <grp> is the subgroup generated by all minimal normal soluble subgroups of <grp>. \>SocleComponents(<grp>) A This function returns a list of minimal normal subgroups of <grp> such that the socle of <grp> (see "Socle") is the direct product of these minimal normal subgroups. Note that, in general, this decomposition is not unique. Currently, this function is only implemented for finite soluble groups. See also "SolubleSocleComponents" and "PSocleComponents". \>AbelianSocleComponents(<grp>) A \>SolubleSocleComponents(<grp>) A \>SolvableSocleComponents(<grp>) A This function returns a list of soluble minimal normal subgroups of <grp> such that the socle of <grp> (see "Socle") is the direct product of these minimal normal subgroups. Note that, in general, this decomposition is not unique. \>PSocle(<grp>, <p>) A If $p$ is a prime, the $p$-socle of a group <grp> is the subgroup generated by all minimal normal $p$-subgroups of <grp>. \>PSocleComponents(<grp>, <p>) A For a prime $p$, this function returns a list of minimal normal $p$-subgroups of <grp> such that the $p$-socle of <grp> (see "PSocle") is the direct product of these minimal normal subgroups. Note that, in general, this decomposition is not unique. \>PSocleSeries(<grp>, <p>) A For a prime <p>, this function returns an ascending <grp>-composition series of the <p>-socle of <grp>. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E %% ¤ Dauer der Verarbeitung: 0.18 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28