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 properties
VIEW RELATED: Group property satisfactions |
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.
