Dear gap-forum, dear Claude,
Does anyone have references with detailled proofs about these methods
and in general for methods to compute \Phi(G).
As you have seen, my article 'Special presentations of finite soluble
groups and computing (pre-) Frattini subgroups' published in the 'Groups
and Computation II' Proceedings in the AMS DIMACS series contains an
effective method to compute the Frattini subgroup of a finite soluble
Is there a specific method for general permutation group ?
I am sure that there are various ideas that you can use to compute the
Frattini subgroup of a finite group G. For example, one can use its
definition as the intersection of all maximal subgroups. But I think
that the following would be more effective:
It is known that Phi(G) is nilpotent. Thus Phi(G) <= F where F is the
Fitting subgroup of G. This yields that Phi(G) can be described as the
intersection of F with those maximal subgroups M of G which do not
contain F. Each such maximal subgroup M is a complement to a G-chief
factor of the form F/N. As F is nilpotent, the G-chief factors F/N in F
can be computed resonably easy and they are abelian. Hence complements
can be determined using the first cohomology group.
I am not aware of any specific reference to a Frattini subgroup algorithm
for finite groups. There are new methods to determine the subgroup lattice
and the maximal subgroups of a permutation group by Cannon and Holt which
are of interest in this context.
Best wishes, Bettina