Quasisimple group

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
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.

Stronger properties

 * Weaker than::Simple non-Abelian group

Weaker properties

 * Stronger than::Inner-simple group: A group whose inner automorphism group is simple
 * Stronger than::Perfect group
 * Stronger than::Group in which every proper normal subgroup is central:
 * Stronger than::Group in which every normal subgroup is a central factor
 * Stronger than::T-group
 * Stronger than::Directly indecomposable group
 * Stronger than::Splitting-simple group
 * Group in which every endomorphism is trivial or an automorphism is weaker than the property of being a finite quasisimple group.

Facts

 * The derived subgroup of an inner-simple group is quasisimple
 * Any normal subgroup of a quasisimple group is either the whole group, or is contained inside the center

Textbook references

 * , Page 156 (definition in paragraph)