LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Frattini subgroup consists exactly of the nongenerators of a finite group

Statement

For a finite group $G$ , an element $x$ lies in $Φ (G)$ if and only if, for every subset $S \subseteq G$ , $⟨ S, x ⟩ = G$ implies $⟨ S ⟩ = G$ . See The Frattini subgroup $Φ (G)$ as the intersection of the maximal subgroups of a finite group.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

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

[L2]

For a subset $S$ of a group $G$ , the generated subgroup is $⟨ S ⟩ : = ⋂ {H : H \leq G and S \subseteq H},$ the smallest subgroup of $G$ containing $S$ . (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

Proof

technique · direct

1.1L1L2givenalgebra

If $x \notin Φ (G)$ , a maximal subgroup omitting $x$ shows that adjoining $x$ can enlarge a generating set.

2.1step 1.1givenalgebra

Conversely, if $⟨ S, x ⟩ = G$ while $⟨ S ⟩ \neq G$ , extend the latter finite subgroup to a maximal subgroup; it contains $S$ but cannot contain $x$ .

3.1step 2.1givenalgebra∎

We treat the trivial group, whose empty intersection convention gives $Φ (1) = 1$ . This proves the stated claim.

Depends on

Used by

Generation of a finite group is detected modulo its Frattini subgroup Corollary

Dependency tree · two levels

6 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)