Frattini's argument

From Groupprops

This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic

This article gives a proof/explanation of the equivalence of multiple definitions for the term [[fact about::automorph-conjugate subgroup]]
View a complete list of pages giving proofs of equivalence of definitions

Statement

Let $H$ be a normal subgroup of $G$ and $P$ an automorph-conjugate subgroup of $H$ . Then:

$H N G (P) = G$

where $N G (P)$ denotes the normalizer of $P$ in $G$ .

Corollaries

We know that any Sylow subgroup of a group is automorph-conjugate, in the sense that any automorphism maps a Sylow subgroup to a conjugate. Hence, in the above statement, we can replace automorph-conjugate subgroup by Sylow subgroup (in fact, that is the more typical way the statement is written).

Proof

Let $g \in G$ . Consider $g P g - 1$ . Since $H$ is normal, the map $x \mapsto gxg^{-1}$ is an automorphism restricted to $H$ . Since $P$ is automorph-conjugate in $H$ , there exists $h \in H$ such that $h P h - 1 = g P g - 1$ .