 FrattiniSubgroup - Maple Help

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GroupTheory

 FrattiniSubgroup
 construct the Frattini subgroup of a group Calling Sequence FrattiniSubgroup( G ) Parameters

 G - a group Description

 • The Frattini subgroup of a group $G$ is the intersection of the maximal subgroups of $G$, or $G$ itself in case $G$ has no maximal subgroups.
 • The Frattini subgroup is equal to 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 a finite group is nilpotent.
 • The FrattiniSubgroup( G ) command returns the Frattini subgroup of a group G. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SmallGroup}\left(32,5\right):$
 > $F≔\mathrm{FrattiniSubgroup}\left(G\right)$
 ${F}{≔}{\Phi }{}\left({\mathrm{< a permutation group on 32 letters with 5 generators >}}\right)$ (1)
 > $\mathrm{GroupOrder}\left(F\right)$
 ${8}$ (2)
 > $\mathrm{IsNilpotent}\left(F\right)$
 ${\mathrm{true}}$ (3)
 > $F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 ${F}{≔}{\Phi }{}\left({{\mathbf{D}}}_{{12}}\right)$ (4)
 > $\mathrm{GroupOrder}\left(F\right)$
 ${2}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{Alt}\left(4\right)\right)\right)$
 ${1}$ (6)

Since a quasicyclic group has no maximal subgroups, it is equal to its Frattini subgroup.

 > $G≔\mathrm{QuasicyclicGroup}\left(7\right)$
 ${G}{≔}{{ℤ}}_{{{7}}^{{\mathrm{\infty }}}}$ (7)
 > $\mathrm{FrattiniSubgroup}\left(G\right)$
 ${{ℤ}}_{{{7}}^{{\mathrm{\infty }}}}$ (8)
 > $F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DirectProduct}\left(\mathrm{SemiDihedralGroup}\left(6\right),\mathrm{GL}\left(2,3\right)\right)\right)$
 ${F}{≔}{\Phi }{}\left({\mathrm{< a permutation group on 32 letters with 4 generators >}}\right)$ (9)
 > $\mathrm{AreIsomorphic}\left(F,\mathrm{DirectProduct}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{SemiDihedralGroup}\left(6\right)\right),\mathrm{FrattiniSubgroup}\left(\mathrm{GL}\left(2,3\right)\right)\right)\right)$
 ${\mathrm{true}}$ (10) Compatibility

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