From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Frattini argument
Date: 1 Dec 2000 10:47:36 GMT
Newsgroups: sci.math
Summary: [missing]

In article <067mlpvy13tf@forum.mathforum.com>,
	Brba12@polaris.com (Brad Baker) writes:
>In connection with group theory I read about the term 
>"Frattini argument". can someone explain what it is ?
>
>Thanks,
> Brad

Theorem - Let N be a normal subgroup of a finite group G, and let P
be a Sylow p-subgroup of N. Then G = N_G(P) N.
(Here (N_G(P) is the normalizer in G of P.)

Proof. Let g in G. Then P^g (= g^{-1}Pg) is contained in N by normality
of N, so P^g is a Sylow p-subgroup of N and hence conjugate to P in N,
by Sylow's Theorem. Thus there exists h in N with P^g = P^h, buth then
gh^{-1} is in N_G(P) so g = (gh^{-1})h is in N_G(P) N.  QED.

This is the original Frattini argument. Other arguments similar to this,
are also sometimes referred to by the same term.

Derek Holt.