Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-06
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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 generalized Fitting subgroup contains its centralizer

Statement

For a finite group G, CG(F∗(G))≤F∗(G).

Facts & Assumptions

Given: Let G be finite and put F∗(G)=F(G)E(G).

[L1]

Smith's generalized-Fitting theorem says that F(G)E(G) contains its centralizer in every finite group G.

Proof

technique · direct
1.1

Smith's generalized-Fitting theorem states, with these conventions, that the product of the Fitting subgroup and the layer is self-centralizing. Its component input is Distinct components commute, and its finite-group hypotheses are exactly those in the statement.

L1given
2.1

Therefore CG(F(G)E(G))≤F(G)E(G), which is CG(F∗(G))≤F∗(G).

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

13 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