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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

group.

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

Miles-Receive-Header: reply

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