# Frobenius group

From Groupprops

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)

View other properties of finite groups OR View all group properties

## Definition

A **Frobenius group** is a finite group satisfying the following equivalent conditions:

- It possesses a Frobenius subgroup: a proper, nontrivial malnormal subgroup.
- It possesses a Frobenius kernel: a proper nontrivial complemented normal centrally closed subgroup.
- It can be expressed as the internal semidirect product of a Frobenius kernel (a centrally closed normal subgroup) and a Frobenius subgroup (a proper nontrivial malnormal subgroup).

## Relation with other properties

### Stronger properties

- Dihedral group of odd degree: A dihedral group where is odd.
- General affine group:GA(1,q)