Quasisimple group

In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension "E" of a simple group "S". In other words, there is a short exact sequence

:1 → "Z"("E") → "E" → "S" → 1

such that "E" = ["E", "E"] where "Z"("E") denotes the center of "E" and [ , ] denotes the commutator. Equivalently, a group is quasi-simple if its inner automorphism group is simple. More simply, a perfect group is quasi-simple if its quotient by its center is simple.

The subnormal quasi-simple subgroups of a group control the structure of a finite insoluble group in much the same way as the minimal normal subgroups of a finite soluble group do, and so are given a name, component. The subgroup generated by the subnormal quasi-simple subgroups is called the layer, and along with the minimal normal soluble subgroups generates a subgroup called the generalized Fitting subgroup. The quasi-simple groups are often studied alongside the simple groups and groups related to their automorphism groups, the almost-simple groups. The representation theory of the quasi-simple groups is nearly identical to the projective representation theory of the simple groups.

ee also

* Schur multiplier

References

*Aschbacher, Michael: "Finite Group Theory", Cambridge University Press, 2000, ISBN 0-521-78675-4