CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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A finite $p$ -group has trivial Frattini subgroup exactly when it is elementary abelian

Statement

A finite $p$ -group has trivial Frattini subgroup if and only if it is elementary abelian.

Facts & Assumptions

Given: A finite $p$ -group $P$ .

[L1]

For a finite $p$ -group $P$ , $P / Φ (P)$ is elementary abelian, and $P / N$ is elementary abelian exactly when $Φ (P) \leq N$ (The Frattini quotient is the largest elementary abelian quotient of a finite $p$ -group).

[F1]

An elementary abelian $p$ -group is a finite abelian $p$ -group in which every nonidentity element has order $p$ ; the trivial group is permitted (Elementary abelian $p$ -groups).

Proof

technique · direct

1.1

For the forward direction, if $Φ (P) = 1$ , then [L1] identifies $P$ with its elementary abelian Frattini quotient.

givenL1F1

2.1

For the reverse direction, if $P$ is elementary abelian, apply the kernel criterion in [L1] with $N = 1$ to obtain $Φ (P) \leq 1$ , hence $Φ (P) = 1$ .

givenL1F1∎

Depends on

Used by

The Frattini subgroup of (ℤ/p)ⁿ is trivial Example

Dependency tree · two levels

16 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, Proposition 2.24 (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, Lemma 3.4 (standard reference, not scraped)

A finite pp-group has trivial Frattini subgroup exactly when it is elementary abelian