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practice wrote:
> Can anyone help?
>
> H is a normal subgroup of G, for G finite. A prime p divides the order
> of H, let P be a Sylow p-subgroup of K. Show that G = H*N_G(P) where
> N_G(P) is the normalizer of P in G.
>
> Thanks.
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Hi:
Imo, this is once of the most beautiful results in basic group theory,
and very simple to prove. Google up "Frattini's Argument"...but only
after you really tried to prove it by yourself (hint: take any g in G
--> P^g is then in H (why?) and thus a Sylow p-sbgp. of H ==> it must
be conjugated to P within H....etc.).
Regards
Tonio
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