Almost simple group

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

• There is a simple non-abelian group $S$ such that $S\le T\le \mathrm{A}\mathrm{u}\mathrm{t}(S)$ 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.

Relation with other properties

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.