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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Every element outside $Φ (P)$ belongs to a minimal generating set of $P$

Statement

Every element of $P ∖ Φ (P)$ belongs to a minimal generating set of the finite $p$ -group $P$ .

Facts & Assumptions

Given: A finite $p$ -group $P$ and $x \in P ∖ Φ (P)$ (The Frattini subgroup $Φ (G)$ as the intersection of the maximal subgroups of a finite group).

[L1]

A subset $X \subseteq P$ is minimally generating exactly when the quotient map restricts to a bijection from $X$ onto a basis of $P / Φ (P)$ (Burnside Basis Theorem).

[L2]

Every independent subset of a finite elementary abelian $p$ -group extends to a basis (Finite elementary abelian $p$ -groups have bases, basis extension, and a well-defined dimension).

[L3]

For a finite $p$ -group $P$ , the quotient $P / Φ (P)$ is elementary abelian (The Frattini quotient is the largest elementary abelian quotient of a finite $p$ -group).

Proof

technique · direct

1.1

By [L3] the quotient $P / Φ (P)$ is a finite elementary abelian $p$ -group, so [L2] applies to it. The coset $x Φ (P)$ is nonzero, so its singleton is independent in $P / Φ (P)$ .

givenL2L3algebra

2.1

Extend that singleton by [L2] to a finite basis. For every other basis vector choose one lift in $P$ and adjoin it to $x$ . The quotient map restricts to a bijection from this lifted set onto the basis, so [L1] makes it a minimal generating set containing $x$ . The assertion is vacuous for $P = 1$ .

step 1.1L1L2given∎

Depends on

Used by

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Dependency tree · two levels

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Sources

M. van Beek, Topics in Finite p-Groups, remark after Theorem 3.7 (standard reference, not scraped)

Every element outside Φ(P)Φ(P) belongs to a minimal generating set of PP