# Group with two conjugacy classes

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **group with two conjugacy classes** is a nontrivial group satisfying the following equivalent conditions:

- All its non-identity element are conjugate.
- The inner automorphism group acts transitively on the set of non-identity elements.
- It has exactly two conjugacy classes of elements.

## Relation with other properties

### Stronger properties

### Weaker properties

- Simple group
- Group in which every element is order-conjugate
- Rational group
- Ambivalent group
- Group having a class-inverting automorphism
- Group whose automorphism group is transitive on non-identity elements
- Group in which every element is order-automorphic
- Group in which any two elements generating the same cyclic subgroup are automorphic
- Group in which every element is automorphic to its inverse