Quasisimple group

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 implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Quasisimple group, all facts related to Quasisimple group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

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 $G$).

Relation with other properties

Stronger properties

• simple non-Abelian group

Weaker properties

• Inner-simple group: A group whose inner automorphism group is simple

Facts

• The commutator 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 For full proof, refer: Normal in quasisimple implies central

References

Textbook references

• Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754^{More info}, Page 156 (definition in paragraph)