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False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24
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FALSE: the Frattini subgroup is the union of the maximal subgroups

Statement

False claim. For a finite group G, the Frattini subgroup Φ(G) is the union of all maximal proper subgroups of G.

Facts & Assumptions

Given: The elementary abelian group E=(Z/2)2.

[L1]

For every prime p and n∈N, Φ((Z/p)n)=1 and the generator rank is n (The Frattini subgroup of (Z/p)n is trivial).

[F1]

The Frattini subgroup is the intersection of the maximal proper subgroups (The Frattini subgroup Φ(G) as the intersection of the maximal subgroups of a finite group, Maximal proper subgroups).

Refutation

technique · direct
1.1

Every nonzero vector of E spans one of the three order-two subgroups ⟨(1,0)⟩, ⟨(0,1)⟩, and ⟨(1,1)⟩. Each has index two and is maximal.

givenL1algebra
2.1

Their union is all of E, while their intersection is {(0,0)}, which is Φ(E) by [L1] and [F1]. Since E is nontrivial, the union is not the Frattini subgroup.

step 1.1L1F1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources