# Almost simple group

From Groupprops

## Contents

## Definition

### Symbol-free definition

A group is said to be **almost simple** if it satisfies the following equivalent conditions:

- There is a simple non-abelian group such that the given group can be embedded between the simple group and its automorphism group.
- The group has a centralizer-free non-abelian simple normal subgroup.

### Definition with symbols

A group is said to be **almost simple** if it satisfies the following equivalent conditions:

- There is a simple non-abelian group such that for some group isomorphic to .
- There exists a normal subgroup of such that is a simple non-abelian group and is trivial.

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

This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

## Relation with other properties

### Stronger properties

## Facts

- Automorphism group of simple non-Abelian group is complete
- Almost simple not implies simple or complete: An almost simple group need not be either simple or complete: in other words, it can be properly sandwiched between a simple group and its automorphism group.
- Symmetric groups are almost simple: For , the symmetric group on letters is almost simple. Note that for , it is in fact the whole automorphism group.