Almost simple group

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 $$G$$ is said to be almost simple if it satisfies the following equivalent conditions:


 * There is a simple non-abelian group $$S$$ such that $$S \le T \le \operatorname{Aut}(S)$$ for some group $$T$$ isomorphic to $$G$$.
 * There exists a normal subgroup $$N$$ of $$G$$ such that $$N$$ is a simple non-abelian group and $$C_G(N)$$ is trivial.

Stronger properties

 * Simple non-Abelian group

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 $$n \ge 5$$, the symmetric group on $$n$$ letters is almost simple. Note that for $$n \ne 6$$, it is in fact the whole automorphism group.