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GroupTheory

 FrattiniSubgroup
 construct the Frattini subgroup of a group

 Calling Sequence FrattiniSubgroup( G )

Parameters

 G - a permutation group

Description

 • The Frattini subgroup of a finite group $G$ is the set of "non-generators" of $G$.  An element $g$ of $G$ is a non-generator if, whenever $G$ is generated by a set $S$ containing $g$, it is also generated by $S\setminus \left\{g\right\}$.
 • The Frattini subgroup of $G$ is also equal to the intersection of the maximal subgroups of $G$. The Frattini subgroup of a finite group is nilpotent.
 • The FrattiniSubgroup( G ) command returns the Frattini subgroup of a group G. The group G must be an instance of a permutation group.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SmallGroup}\left(32,5\right):$
 > $F≔\mathrm{FrattiniSubgroup}\left(G\right)$
 ${F}{≔}{\Phi }{}\left(⟨\left({1}{,}{2}{,}{6}{,}{11}{,}{8}{,}{12}{,}{7}{,}{3}\right)\left({4}{,}{15}{,}{18}{,}{30}{,}{20}{,}{31}{,}{19}{,}{16}\right)\left({5}{,}{10}{,}{21}{,}{27}{,}{23}{,}{28}{,}{22}{,}{14}\right)\left({9}{,}{24}{,}{25}{,}{32}{,}{26}{,}{29}{,}{13}{,}{17}\right){,}\left({1}{,}{4}\right)\left({2}{,}{9}\right)\left({3}{,}{13}\right)\left({5}{,}{17}\right)\left({6}{,}{18}\right)\left({7}{,}{19}\right)\left({8}{,}{20}\right)\left({10}{,}{15}\right)\left({11}{,}{25}\right)\left({12}{,}{26}\right)\left({14}{,}{16}\right)\left({21}{,}{24}\right)\left({22}{,}{29}\right)\left({23}{,}{32}\right)\left({27}{,}{30}\right)\left({28}{,}{31}\right){,}\left({1}{,}{5}\right)\left({2}{,}{10}\right)\left({3}{,}{14}\right)\left({4}{,}{17}\right)\left({6}{,}{21}\right)\left({7}{,}{22}\right)\left({8}{,}{23}\right)\left({9}{,}{15}\right)\left({11}{,}{27}\right)\left({12}{,}{28}\right)\left({13}{,}{16}\right)\left({18}{,}{24}\right)\left({19}{,}{29}\right)\left({20}{,}{32}\right)\left({25}{,}{30}\right)\left({26}{,}{31}\right){,}\left({1}{,}{6}{,}{8}{,}{7}\right)\left({2}{,}{11}{,}{12}{,}{3}\right)\left({4}{,}{18}{,}{20}{,}{19}\right)\left({5}{,}{21}{,}{23}{,}{22}\right)\left({9}{,}{25}{,}{26}{,}{13}\right)\left({10}{,}{27}{,}{28}{,}{14}\right)\left({15}{,}{30}{,}{31}{,}{16}\right)\left({17}{,}{24}{,}{32}{,}{29}\right){,}\left({1}{,}{8}\right)\left({2}{,}{12}\right)\left({3}{,}{11}\right)\left({4}{,}{20}\right)\left({5}{,}{23}\right)\left({6}{,}{7}\right)\left({9}{,}{26}\right)\left({10}{,}{28}\right)\left({13}{,}{25}\right)\left({14}{,}{27}\right)\left({15}{,}{31}\right)\left({16}{,}{30}\right)\left({17}{,}{32}\right)\left({18}{,}{19}\right)\left({21}{,}{22}\right)\left({24}{,}{29}\right)⟩\right)$ (1)
 > $\mathrm{GroupOrder}\left(F\right)$
 ${8}$ (2)
 > $\mathrm{IsNilpotent}\left(F\right)$
 ${\mathrm{true}}$ (3)
 > $F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 ${F}{≔}{\Phi }{}\left({{\mathrm{D}}}_{{12}}\right)$ (4)
 > $\mathrm{GroupOrder}\left(F\right)$
 ${2}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{Alt}\left(4\right)\right)\right)$
 ${1}$ (6)

Compatibility

 • The GroupTheory[FrattiniSubgroup] command was introduced in Maple 17.