TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The Frattini subgroup of a finite group is nilpotent

Statement

The Frattini subgroup of every finite group is nilpotent. See Nilpotence lifts over the Frattini subgroup of a finite group.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

Let $G$ be finite and let $Φ (G) \leq N ⊴ G$ . Then $N$ is nilpotent if and only if $N / Φ (G)$ is nilpotent. In particular, $G$ is nilpotent if and only if $G / Φ (G)$ is nilpotent. (Nilpotence lifts over the Frattini subgroup of a finite group).

Proof

technique · direct

1.1L1givenalgebra

Every automorphism of $G$ permutes the maximal proper subgroups, so their intersection $Φ (G)$ is characteristic and therefore normal; the lifting theorem applies with $N = Φ (G)$ , whose quotient by $Φ (G)$ is the trivial nilpotent group.

2.1step 1.1givenalgebra∎

The trivial group is admitted: it has no maximal proper subgroup, so the defining family is empty and its intersection inside $G$ is $G$ itself, giving $Φ (1) = 1$ , which is nilpotent of class zero. This proves the stated claim.

Depends on

Nilpotence lifts over the Frattini subgroup of a finite group

Used by

Dependency tree · two levels

8 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

David A. Craven, Finite Group Theory, Sections 1.4 and 2.3 (standard reference, not scraped)