DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

The Fitting subgroup $F (G) = \prod_{p} O_{p} (G)$ of a finite group

Definition

For a finite group $G$ , the Fitting subgroup is $F (G) : = \prod_{p ∣ ∣ G ∣} O_{p} (G),$ the product of its $p$ -cores (The $p$ -core $O_{p} (G)$ as the largest normal $p$ -subgroup). If $A, B ⊴ G$ , then $A B$ is a subgroup, is normal because $g A B g^{- 1} = A B$ , and satisfies $A B = B A$ : indeed $a b = (a b a^{- 1}) a \in B A$ , and the reverse inclusion is symmetric. Induction therefore shows that the finite product of the normal factors $O_{p} (G)$ is a normal subgroup independent of their order. For the trivial group the product is empty and equals $1$ .

Depends on

Used by

Dependency tree · two levels

8 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

Rachel K. Carleton, The Commuting and Cyclic Graphs of Solvable A-Groups, Chapter 2 Section 2.1 (standard reference, not scraped)

The Fitting subgroup F(G)=∏pOp(G)F(G)=∏p​Op​(G) of a finite group