---
tag: Z20AM1KT9R
kind: definition
name: Frobenius group
msc: [20B05, 20D99]
defines: JXS91FGFHY
uses: [0WCY42EV9V, 02GK6HZYBT, 0X1JXT47GA, 0H11ZPX4ND, 0ZXZ82FTX2, 0JZ27DMJ5D, XK7H3YXBQT]
sources:
  - url: https://en.wikipedia.org/wiki/Frobenius_group
---

## Definition

Let a finite group $G$ act [[transitively|02GK6HZYBT]] on a finite set $\Omega$ with $\abs{\Omega} \ge 2$.
$G$ is a *Frobenius group* (for this action) if:

1. no element $g \ne e$ of $G$ fixes two distinct points of $\Omega$, and
2. some element $g \ne e$ of $G$ fixes a point of $\Omega$.

## Remarks

Equivalently, a finite group $G$ is a Frobenius group if it has a [[subgroup|0X1JXT47GA]] $H$ with $\set{e} \ne H \ne G$ and $H \cap gHg^{-1} = \set{e}$ for all $g \in G \setminus H$.
Then $H$ is a point stabiliser for the action on the left cosets of $H$, and it is called a *Frobenius complement*.
The set $K$ of elements that fix no point, together with $e$, is the *Frobenius kernel*; by [[Frobenius's theorem|XK7H3YXBQT]] it is a normal subgroup.

Example: the group of maps $x \mapsto ax + b$ with $a \ne 0$, acting on a [[finite field|0ZXZ82FTX2]] $\F_q$ with $q \ge 3$.
If $a \ne 1$, the equation $ax + b = x$ has exactly one solution, and if $a = 1$ and $b \ne 0$ it has none, so clause 1 holds.
The map $x \mapsto ax$ with $a \ne 0, 1$ fixes $0$, so clause 2 holds.
For $q = 3$ this is $S_3$ acting on $3$ points.
For odd $n \ge 3$, the [[dihedral group|0JZ27DMJ5D]] of order $2n$ acting on the corners of a regular $n$-gon is also a Frobenius group.

Non-example: $S_4$ acting on $\set{1, 2, 3, 4}$.
The transposition $(1\,2)$ fixes the two points $3$ and $4$, so clause 1 fails.
The cyclic group generated by $(1\,2\,3)$ acting on $\set{1, 2, 3}$ satisfies clause 1 but fails clause 2.