Frattini's argument

Frattini's argument

Statement

If G is a finite group with normal subgroup H, and if P is a Sylow p-subgroup of H, then

: G = N_G(P)H,

where N_G(P) denotes the normalizer of P in G, and N_G(P)H means the product of group subsets.

Proof

The group P is a Sylow p-subgroup of H, so every Sylow p-subgroup of H is an H-conjugate of P, that is, it is of the form h^{-1}Ph for some h \in H (see Sylow theorems). Let g be any element of G. Since H is normal in G, the subgroup g^{-1}Pg is contained in H. This means that g^{-1}Pg is a Sylow p-subgroup of H. Then, by the above, it must be H-conjugate to P: that is, for some h \in H

: g^{-1}Pg = h^{-1}Ph,

and so

: hg^{-1}Pgh^{-1} = P.

Thus

: gh^{-1} \in N_G(P),

and therefore g \in N_G(P)H. But g \in G was arbitrary, and so G = HN_G(P) = N_G(P)H.\ \square

Applications

• Frattini's argument can be used as part of a proof that any finite nilpotent group is a direct product of its Sylow subgroups.
• By applying Frattini's argument to N_G(N_G(P)), it can be shown that N_G(N_G(P)) = N_G(P) whenever G is a finite group and P is a Sylow p-subgroup of G.
• More generally, if a subgroup M \leq G contains N_G(P) for some Sylow p-subgroup P of G, then M is self-normalizing, i.e. M = N_G(M).

References

References

1. M. Brescia, F. de Giovanni, M. Trombetti, [http://www.advgrouptheory.com/journal/Volumes/3/M.%20Brescia,%20F.%20de%20Giovanni,%20M.%20Trombetti%20-%20The%20true%20story%20behind%20Frattinis%20Argument.pdf "The True Story Behind Frattini’s Argument"], ''[[Advances in Group Theory and Applications]]'' '''3''', [https://doi.org/10.4399/97888255036928 doi:10.4399/97888255036928]