TheoremStatement: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$F (G / Φ (G)) = F (G) / Φ (G)$ for every finite group

Statement

For every finite group $G$ , $F (G / Φ (G)) = F (G) / Φ (G) .$ See The Fitting subgroup is nilpotent and is the largest normal nilpotent subgroup of a finite group.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

For every finite group $G$ , $F (G)$ is nilpotent and normal, and every normal nilpotent subgroup of $G$ is contained in $F (G)$ . (The Fitting subgroup is nilpotent and is the largest normal nilpotent subgroup of a finite group).

[L2]

For every finite group $G$ , $Φ (G) \leq F (G)$ . (The Frattini subgroup is contained in the Fitting subgroup).

[L3]

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).

[L4]

Correspondence theorem: subgroups of $G / N$ correspond to subgroups of $G$ containing $N$ , with normality preserved. For $N ⊴ G$ , the maps $H \mapsto H / N$ and $K \mapsto π^{- 1} (K)$ are inverse inclusion-preserving bijections between subgroups $H$ with $N \leq H \leq G$ and subgroups $K \leq G / N$ ; they preserve normality. (Correspondence theorem: subgroups of $G / N$ correspond to subgroups of $G$ containing $N$ , with normality preserved).

Proof

technique · direct

1.1L1L2L3L4givenalgebra

The image $F (G) / Φ (G)$ is normal and nilpotent, giving one inclusion.

2.1step 1.1givenalgebra

For the reverse inclusion, pull $F (G / Φ (G))$ back to a normal subgroup $N$ of $G$ ; the lifting theorem makes $N$ nilpotent, so $N \leq F (G)$ .

3.1step 1.1step 2.1L2givenalgebra∎

If $G / Φ (G)$ is trivial, then $Φ (G) = G$ , and $Φ (G) \leq F (G) \leq G$ from [L2] forces $F (G) = G$ ; both sides of the identity are then the trivial group. Together with the two inclusions of steps 1.1 and 2.1 this gives equality in every case. This proves the stated claim.

Depends on

Used by

The p-cores, Fitting subgroup, and Frattini subgroup of S₄ Example

Dependency tree · two levels

21 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)

F(G/Φ(G))=F(G)/Φ(G)F(G/Φ(G))=F(G)/Φ(G) for every finite group