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: every homomorphism carries the Frattini subgroup into the target Frattini subgroup

Statement

False claim. Every homomorphism $f : G \to H$ between finite groups satisfies $f (Φ (G)) \leq Φ (H)$ .

Facts & Assumptions

Given: Finite groups and their Frattini subgroups (The Frattini subgroup $Φ (G)$ as the intersection of the maximal subgroups of a finite group).

[L1]

The embedding $C_{4} ↪ S_{5}$ sending a generator to a $4$ -cycle has $f (Φ (C_{4})) ≰ Φ (S_{5})$ (A nonsurjective homomorphism need not carry the Frattini subgroup into the target Frattini subgroup).

Refutation

technique · direct

1.1

The embedding in [L1] is a homomorphism for which the claimed inclusion fails, so it refutes the universal statement.

1.2

Surjectivity gives the valid replacement. If $f : G \to H$ is onto and $M$ is maximal in $H$ , then $f^{- 1} (M)$ is maximal in $G$ : a subgroup $K$ properly containing it has $f (K) > M$ , hence $f (K) = H$ ; for every $g \in G$ , choose $k \in K$ with $f (k) = f (g)$ , and then $g k^{- 1} \in \ker f \leq f^{- 1} (M) \leq K$ , so $K = G$ . Therefore every $x \in Φ (G)$ lies in every $f^{- 1} (M)$ , and $f (x)$ lies in every maximal $M$ , hence in $Φ (H)$ .

givenalgebra

2.1

Step 1.1 shows that nonsurjective homomorphisms need not satisfy the inclusion, while step 1.2 identifies the missing sufficient hypothesis.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, §2.2 (standard reference, not scraped)
J. S. Milne, Group Theory (standard reference, not scraped)