# Quasisimple group

## Definition

### Symbol-free definition

A group is said to be quasisimple if it is perfect and its inner automorphism group is simple.

### Definition with symbols

A group $G$ is said to be quasisimple if both the following hold:

• $G$ is perfect, that is, $G' = G$
• The inner automorphism group of $G$ is a simple group, that is, $G/Z(G)$ is simple (where $Z(G)$ denotes the center of $G$).

## Classification

Further information: Classification of finite simple groups

The finite quasisimple groups can be completely classified in terms of the finite simple non-abelian groups. Specifically, for each finite simple non-abelian group $S$, consider the Schur multiplier and the corresponding Schur covering group (the unique universal central extension) $\hat S$ with its map to $G$. The quasisimple groups with inner automorphism group $S$ are precisey the groups $S$ such that the map $\hat S \to S$ can be factored in terms of a surjective map $\hat S \to G$ and a surjective map $G \to S$.

In particular, by the fourth isomorphism theorem, these correspond precisely to the quotient groups (and hence to the subgroups, because of abelianness) of $\hat S/S$, which is the Schur multiplier of $S$. In particular, for each finite simple non-abelian groups, there are finitely many quasisimple groups associated with it.