Quasisimple group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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
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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:

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

Weaker properties

Facts

References

Textbook references

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