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The Frattini subgroup of a finite group is characteristic

Statement

For every finite group $G$ , the subgroup $Φ (G)$ is characteristic in $G$ and hence normal.

Facts & Assumptions

Given: A finite group $G$ and an automorphism $α$ of $G$ .

[F1]

For a finite group $G$ , the Frattini subgroup is $Φ (G) = ⋂ {M \leq G : M is maximal proper}$ ; if $G = 1$ , the empty intersection inside $G$ is $G$ (The Frattini subgroup $Φ (G)$ as the intersection of the maximal subgroups of a finite group).

[F2]

A subgroup $H \leq G$ is characteristic when $α (H) = H$ for every automorphism $α$ of $G$ (Characteristic subgroups).

Proof

technique · direct

1.1

If $M$ is maximal proper, then $α (M)$ is proper and maximal: any subgroup strictly between $α (M)$ and $G$ pulls back under $α^{- 1}$ to one strictly between $M$ and $G$ . Thus $α$ permutes the family of maximal proper subgroups.

givenF1algebra

2.1

An automorphism carries an intersection to the intersection of the images, so step 1.1 and [F1] give $α (Φ (G)) = Φ (G)$ . This is characteristicity by [F2]. Every inner automorphism is an automorphism, so $Φ (G)$ is normal. The same argument covers $G = 1$ .

step 1.1F1F2algebra∎

Depends on

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. A. Craven, The Theory of p-Groups, §2.2 (standard reference, not scraped)
K. Conrad, Generating Sets, §6 (standard reference, not scraped)