construct the Frattini series of a group
return the Frattini length of a group
FrattiniSeries( G )
FrattiniLength( G )
a permutation group
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 G0=G and, for 0<k, define Gk=Φ⁡Gk−1. 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.
G ≔ DihedralGroup⁡8:
fs ≔ FrattiniSeries⁡G
G ≔ ASL⁡2,3
G ≔ FrobeniusGroup⁡18,1
The GroupTheory[FrattiniSeries] and GroupTheory[FrattiniLength] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
Download Help Document