Quasisimple group

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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.

Examples

Finite simple non-abelian group	Order	Schur multiplier	Schur covering group	Quasisimple groups with this simple group as the inner automorphism group
alternating group:A5	60	cyclic group:Z2	special linear group:SL(2,5)	alternating group:A5, special linear group:SL(2,5)
projective special linear group:PSL(3,2) (isomorphic to $PSL(2,7)$)	168	cyclic group:Z2	special linear group:SL(2,7)	projective special linear group:PSL(3,2), special linear group:SL(2,7)
alternating group:A6	360	cyclic group:Z6	Schur cover of alternating group:A6	alternating group:A6, special linear group:SL(2,9), triple cover of alternating group:A6, Schur cover of alternating group:A6
projective special linear group:PSL(2,8)	504	trivial group	projective special linear group:PSL(2,8)	projective special linear group:PSL(2,8)
projective special linear group:PSL(2,11)	660	cyclic group:Z2	special linear group:SL(2,11)	projective special linear group:PSL(2,11), special linear group:SL(2,11)

Relation with other properties

Stronger properties

• Simple non-Abelian group

Weaker properties

• Inner-simple group: A group whose inner automorphism group is simple
• Perfect group
• Group in which every proper normal subgroup is central: For full proof, refer: Proper and normal in quasisimple implies central
• Group in which every normal subgroup is a central factor
• T-group
• Directly indecomposable group
• Splitting-simple group
• Group in which every endomorphism is trivial or an automorphism is weaker than the property of being a finite quasisimple group. For full proof, refer: Finite quasisimple implies every endomorphism is trivial or an automorphism

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 For full proof, refer: Proper and 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)