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CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17
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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.
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The Frattini subgroup is contained in the Fitting subgroup

Statement

For every finite group G, Φ(G)≤F(G). See The Frattini subgroup of a finite group is nilpotent.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

The Frattini subgroup of every finite group is nilpotent. (The Frattini subgroup of a finite group is nilpotent).

[L2]

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).

Proof

technique · direct
1.1

The Frattini subgroup is characteristic, hence normal, and is nilpotent; maximality of the Fitting subgroup gives the inclusion.

L1L2givenalgebra
2.1

No finiteness beyond that of the Statement is used, and the degenerate case is consistent: for G={1} the family of maximal proper subgroups is empty, so Φ(G)={1}=F(G) and the inclusion holds with equality. This proves the stated claim.

step 1.1givenalgebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources