---
tag: 7Q3X4FG7A5
kind: definition
name: Fitting subgroup
msc: [20D25]
defines: N219CDB623
uses: [G5S1M9, 0CB0DX5BVV, 0Q7MMQHC1P, 0RP7VWQ0YN, 0KABPMRHM2, 0FXJ7YFMMX, 02YYV36BFM, 0PMRQJVTME, 0DZ93493VN, 0YMYMNQ0QZ, 0053CQE9Y2]
sources:
  - url: https://en.wikipedia.org/wiki/Fitting_subgroup
---

## Definition

Let $G$ be a finite [[group|G5S1M9]].
The *Fitting subgroup* $F(G)$ is the subgroup [[generated|0RP7VWQ0YN]] by all [[normal subgroups|0CB0DX5BVV]] of $G$ that are [[nilpotent|0Q7MMQHC1P]].

## Remarks

Fitting's theorem says that if $M$ and $N$ are nilpotent normal subgroups of $G$, then $MN$ is a nilpotent normal subgroup.
Hence $F(G)$ is itself nilpotent and normal, and it contains every nilpotent normal subgroup of $G$: it is the unique largest one.
$F(G)$ is a [[characteristic subgroup|0KABPMRHM2]], and it is the product over the primes $p$ of the largest normal $p$-subgroups $O_p(G)$.
If $G$ is [[solvable|0FXJ7YFMMX]], then the [[centraliser|02YYV36BFM]] of $F(G)$ in $G$ is contained in $F(G)$.

Example: $G = S_4$.
Its normal subgroups are $\set{e}$, $V = \set{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)}$, $A_4$ and $S_4$.
The group $V$ is abelian, hence nilpotent, and it is the [[Klein four-group|0PMRQJVTME]].
The groups $A_4$ and $S_4$ are not nilpotent, because their [[centres|0DZ93493VN]] are trivial.
So $F(S_4) = V$.

Non-example: $A_4$ is a normal subgroup of $S_4$ that is not nilpotent, so it is not contained in the generating family, and it is not $F(S_4)$.

If $G$ is a nonabelian [[simple group|0YMYMNQ0QZ]], then $F(G) = \set{e}$.
If $G$ is nilpotent, then $F(G) = G$.