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Automorphisms act linearly on the Frattini quotient

Statement

Every automorphism of a finite $p$ -group induces an $F_{p}$ -linear automorphism of $P / Φ (P)$ , and these form a homomorphism

$ρ_{P} : Aut (P) ⟶ {Aut}_{F_{p}} (P / Φ (P)) .$

Facts & Assumptions

Given: A finite $p$ -group $P$ , an automorphism $α \in Aut (P)$ , and the quotient map $π : P \to P / Φ (P)$ .

[L1]

For every finite group $G$ , $Φ (G)$ is characteristic and hence normal (The Frattini subgroup of a finite group is characteristic).

[L2]

The rule $\overset{ˉ}{a} \cdot x = x^{a}$ gives every elementary abelian $p$ -group its canonical $F_{p}$ -vector-space structure (An elementary abelian $p$ -group has a canonical $F_{p}$ -vector-space structure, The Frattini quotient is the largest elementary abelian quotient of a finite $p$ -group).

[L3]

The automorphisms of a group form a group under composition (The automorphisms of a group form a group under composition).

[L4]

A homomorphism that kills a normal subgroup factors uniquely through the quotient group (A homomorphism that kills a normal subgroup factors uniquely through the quotient group).

Proof

technique · direct

1.1

By [L1], $α (Φ (P)) = Φ (P)$ , so $π \circ α$ kills $Φ (P)$ . The universal property [L4] gives a quotient automorphism $\overset{ˉ}{α}$ with $\overset{ˉ}{α} (x Φ (P)) = α (x) Φ (P)$ ; applying the same construction to $α^{- 1}$ supplies its inverse.

givenL1L4

2.1

For $\overset{ˉ}{a} \in F_{p}$ , $\overset{ˉ}{α} (\overset{ˉ}{a} \cdot x Φ (P)) = \overset{ˉ}{α} (x^{a} Φ (P)) = α (x^{a}) Φ (P) = α (x)^{a} Φ (P) = \overset{ˉ}{a} \cdot \overset{ˉ}{α} (x Φ (P))$ , so $\overset{ˉ}{α}$ is linear by [L2].

step 1.1L2algebra

3.1

The induced map of the identity is the identity, and uniqueness in [L4] gives $\overline{α β} = \overset{ˉ}{α} \overset{ˉ}{β}$ . Thus $α \mapsto \overset{ˉ}{α}$ preserves the group law in [L3] and defines $ρ_{P}$ .

step 1.1step 2.1L3L4algebra∎

Depends on

Used by

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Sources

D. A. Craven, The Theory of p-Groups, discussion before Theorem 2.30 (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, Proposition 4.10 (standard reference, not scraped)