False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-24

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FALSE: the Fitting subgroup always contains its centralizer

Statement

False claim. For every finite group $G$ , one has $C_{G} (F (G)) \leq F (G)$ .

Facts & Assumptions

Given: The Fitting subgroup and its centralizer in a finite group.

[L1]

For $A_{5}$ , one has $F (A_{5}) = 1$ and $C_{A_{5}} (F (A_{5})) = A_{5}$ (The Fitting subgroup of $A_{5}$ does not contain its centralizer).

[L2]

If $G$ is finite and solvable, then $C_{G} (F (G)) \leq F (G)$ (Philip Hall: in a finite solvable group the Fitting subgroup contains its own centralizer).

Refutation

technique · direct

1.1L1

For $A_{5}$ , [L1] gives $F (A_{5}) = 1$ and $C_{A_{5}} (F (A_{5})) = A_{5}$ .

2.1step 1.1L2

Thus the claimed inclusion fails for $A_{5}$ . The valid theorem [L2] has finite solvability as a hypothesis, and $A_{5}$ lies outside that hypothesis.

3.1step 1.1step 2.1∎

The strict failure in step 1.1 refutes the hypothesis-free claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

D. A. Craven, Finite Group Theory, Theorem 2.13 (standard reference, not scraped)