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 $0<k$, define $G_k=\mathrm{\Phi }\left(G_{k-1}\right)$. The sequence

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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$G≔\mathrm{DihedralGroup}\left(8\right)\&colon;$

$\mathrm{fs}≔\mathrm{FrattiniSeries}\left(G\right)$

$\mathrm{fs}≔{\mathbf{D}}_8◃\Phi \left({\mathbf{D}}_8\right)◃\Phi \left(\Phi \left({\mathbf{D}}_8\right)\right)◃\Phi \left(\Phi \left(\Phi \left({\mathbf{D}}_8\right)\right)\right)$

$\mathrm{type}\left(\mathrm{fs}\&comma;'\mathrm{NormalSeries}'\right)$

$\mathrm{true}$

$\mathrm{FrattiniLength}\left(G\right)$

$3$

$:-\mathrm{numelems}\left(\mathrm{fs}\right)$

$4$

$G≔\mathrm{ASL}\left(2\&comma;3\right)$

$G≔\mathbf{ASL}\left(2\&comma;3\right)$

$\mathrm{fs}≔\mathrm{FrattiniSeries}\left(G\right)$

$\mathrm{fs}≔\mathbf{ASL}\left(2\&comma;3\right)◃\Phi \left(\mathbf{ASL}\left(2\&comma;3\right)\right)$

$\mathrm{FrattiniLength}\left(G\right)$

$1$

$\mathrm{numelems}\left(\mathrm{fs}\right)$

$2$

$G≔\mathrm{FrobeniusGroup}\left(18\&comma;1\right)$

$G≔⟨\left(2\&comma;9\right)\left(3\&comma;6\right)\left(4\&comma;8\right)\left(5\&comma;7\right)\&comma;\left(1\&comma;2\&comma;4\&comma;3\&comma;5\&comma;7\&comma;6\&comma;8\&comma;9\right)\&comma;\left(1\&comma;3\&comma;6\right)\left(2\&comma;5\&comma;8\right)\left(4\&comma;7\&comma;9\right)⟩$

$\mathrm{fs}≔\mathrm{FrattiniSeries}\left(G\right)$

$\mathrm{fs}≔⟨\left(2\&comma;9\right)\left(3\&comma;6\right)\left(4\&comma;8\right)\left(5\&comma;7\right)\&comma;\left(1\&comma;2\&comma;4\&comma;3\&comma;5\&comma;7\&comma;6\&comma;8\&comma;9\right)\&comma;\left(1\&comma;3\&comma;6\right)\left(2\&comma;5\&comma;8\right)\left(4\&comma;7\&comma;9\right)⟩◃\Phi \left(⟨\left(2\&comma;9\right)\left(3\&comma;6\right)\left(4\&comma;8\right)\left(5\&comma;7\right)\&comma;\left(1\&comma;2\&comma;4\&comma;3\&comma;5\&comma;7\&comma;6\&comma;8\&comma;9\right)\&comma;\left(1\&comma;3\&comma;6\right)\left(2\&comma;5\&comma;8\right)\left(4\&comma;7\&comma;9\right)⟩\right)◃\Phi \left(\Phi \left(⟨\left(2\&comma;9\right)\left(3\&comma;6\right)\left(4\&comma;8\right)\left(5\&comma;7\right)\&comma;\left(1\&comma;2\&comma;4\&comma;3\&comma;5\&comma;7\&comma;6\&comma;8\&comma;9\right)\&comma;\left(1\&comma;3\&comma;6\right)\left(2\&comma;5\&comma;8\right)\left(4\&comma;7\&comma;9\right)⟩\right)\right)$

$\mathrm{FrattiniLength}\left(G\right)$

$2$

The GroupTheory[FrattiniSeries] and GroupTheory[FrattiniLength] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.