---
tag: N219CDB623
kind: concept
name: Fitting subgroup
msc: [20D25]
specializes: [0KABPMRHM2]
sources:
  - url: https://en.wikipedia.org/wiki/Fitting_subgroup
same_as:
  wikidata: Q3492011
  wikipedia: Fitting subgroup
---

## Summary

The Fitting subgroup $F(G)$ of a finite group $G$ is the largest [[nilpotent|0Q7MMQHC1P]] [[normal subgroup|0CB0DX5BVV]] of $G$.
It exists because the product of two nilpotent normal subgroups is again nilpotent, a theorem of Hans Fitting.
In a [[solvable group|0FXJ7YFMMX]] $G$, the Fitting subgroup contains its own [[centraliser|02YYV36BFM]], so it controls much of the structure of $G$.
It is a basic tool in the theory of finite solvable groups.

## Explanations

### L1 human

Nilpotent groups are the groups that are closest to being abelian.
The Fitting subgroup is the biggest nilpotent piece that sits inside a group as a normal subgroup.
For the group $S_4$ of all permutations of four objects, it is the Klein four-group formed by the identity and the three products of two disjoint swaps.
For a nonabelian simple group, such as $A_5$, the Fitting subgroup is trivial.