---
tag: JXS91FGFHY
kind: concept
name: Frobenius group
msc: [20B05, 20D99]
specializes: [ZDJQRKPWGA]
sources:
  - url: https://en.wikipedia.org/wiki/Frobenius_group
same_as:
  wikidata: Q1457165
  wikipedia: Frobenius group
---

## Summary

A Frobenius group is a finite [[transitive|02GK6HZYBT]] permutation group in which some non-identity element fixes a point, but no non-identity element fixes two points.
The elements that fix no point, together with the identity, form a normal subgroup, the Frobenius kernel, by [[Frobenius's theorem|XK7H3YXBQT]].
The group is then a [[semidirect product|0H3WZXCZNF]] of the kernel and a point [[stabiliser|0AYMP41THN]], the Frobenius complement.
Frobenius groups appear throughout finite group theory and character theory.

## Explanations

### L1 human

Take the three corners of a triangle and all six ways to permute them.
A reflection keeps one corner in place and swaps the other two; a rotation moves every corner.
No symmetry other than the identity keeps two corners in place.
This is the smallest Frobenius group.
Its rotations, together with the identity, form a subgroup, and that is the pattern in every Frobenius group.