GroupTheory/FrattiniSeries - Maple Help

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GroupTheory

 FrattiniSeries
 construct the Frattini series of a group
 FrattiniLength
 return the Frattini length of a group

 Calling Sequence FrattiniSeries( G ) FrattiniLength( G )

Parameters

 G - a permutation group

Description

 • The Frattini series of a group $G$ is the descending normal series of $G$ whose terms are the successive Frattini subgroups, defined as follows. Let ${G}_{0}=G$ and, for $0, define ${G}_{k}=\mathrm{\Phi }\left({G}_{k-1}\right)$. The sequence

$G={G}_{0}▹{G}_{1}▹\dots ▹{G}_{r}$

 of distinct terms is called the Frattini series of $G$. The number $r$ is called the Frattini length of $G$.
 • The FrattiniSeries( G ) command constructs the Frattini series of a group G. The group G must be an instance of a permutation group. The Frattini series of G is represented by a series data structure which admits certain operations common to all series.  See GroupTheory[Series].
 • Since the group G is required to be finite, the Frattini series always terminates in the trivial subgroup.
 • The FrattiniLength( G ) command returns the Frattini length of G; that is, the length of the Frattini series of G. This is the number of subgroup inclusions - so it is one less than the number of groups in the Frattini series.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DihedralGroup}\left(8\right):$
 > $\mathrm{fs}≔\mathrm{FrattiniSeries}\left(G\right)$
 ${\mathrm{fs}}{≔}{{\mathbf{D}}}_{{8}}{▹}{\Phi }{}\left({{\mathbf{D}}}_{{8}}\right){▹}{\Phi }{}\left({\Phi }{}\left({{\mathbf{D}}}_{{8}}\right)\right){▹}{\Phi }{}\left({\Phi }{}\left({\Phi }{}\left({{\mathbf{D}}}_{{8}}\right)\right)\right)$ (1)
 > $\mathrm{type}\left(\mathrm{fs},'\mathrm{NormalSeries}'\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{FrattiniLength}\left(G\right)$
 ${3}$ (3)
 > $:-\mathrm{numelems}\left(\mathrm{fs}\right)$
 ${4}$ (4)
 > $G≔\mathrm{FrobeniusGroup}\left(18,1\right)$
 ${G}{≔}⟨\left({2}{,}{9}\right)\left({3}{,}{6}\right)\left({4}{,}{8}\right)\left({5}{,}{7}\right){,}\left({1}{,}{2}{,}{4}{,}{3}{,}{5}{,}{7}{,}{6}{,}{8}{,}{9}\right){,}\left({1}{,}{3}{,}{6}\right)\left({2}{,}{5}{,}{8}\right)\left({4}{,}{7}{,}{9}\right)⟩$ (5)
 > $\mathrm{fs}≔\mathrm{FrattiniSeries}\left(G\right)$
 ${\mathrm{fs}}{≔}⟨\left({2}{,}{9}\right)\left({3}{,}{6}\right)\left({4}{,}{8}\right)\left({5}{,}{7}\right){,}\left({1}{,}{2}{,}{4}{,}{3}{,}{5}{,}{7}{,}{6}{,}{8}{,}{9}\right){,}\left({1}{,}{3}{,}{6}\right)\left({2}{,}{5}{,}{8}\right)\left({4}{,}{7}{,}{9}\right)⟩{▹}{\Phi }{}\left(⟨\left({2}{,}{9}\right)\left({3}{,}{6}\right)\left({4}{,}{8}\right)\left({5}{,}{7}\right){,}\left({1}{,}{2}{,}{4}{,}{3}{,}{5}{,}{7}{,}{6}{,}{8}{,}{9}\right){,}\left({1}{,}{3}{,}{6}\right)\left({2}{,}{5}{,}{8}\right)\left({4}{,}{7}{,}{9}\right)⟩\right){▹}{\Phi }{}\left({\Phi }{}\left(⟨\left({2}{,}{9}\right)\left({3}{,}{6}\right)\left({4}{,}{8}\right)\left({5}{,}{7}\right){,}\left({1}{,}{2}{,}{4}{,}{3}{,}{5}{,}{7}{,}{6}{,}{8}{,}{9}\right){,}\left({1}{,}{3}{,}{6}\right)\left({2}{,}{5}{,}{8}\right)\left({4}{,}{7}{,}{9}\right)⟩\right)\right)$ (6)
 > $\mathrm{FrattiniLength}\left(G\right)$
 ${2}$ (7)

Compatibility

 • The GroupTheory[FrattiniSeries] and GroupTheory[FrattiniLength] commands were introduced in Maple 2019.
 • For more information on Maple 2019 changes, see Updates in Maple 2019.

 See Also