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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:

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

Relation with other properties

Facts

References

Textbook references

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