DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-17

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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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 7 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

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