Quasisimple group

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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 quasisimple if it is isomorphic to its commutator subgroup and its inner automorphism group Inn(G) (its quotient by its center) is simple; due to Grün's lemma, Inn(G) must be non-abelian. All non-abelian simple groups are quasisimple.

The subnormal quasisimple 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 quasisimple subgroups is called the layer, and along with the minimal normal soluble subgroups generates a subgroup called the generalized Fitting subgroup. The quasisimple groups are often studied alongside the simple groups and groups related to their automorphism groups, the almost-simple groups. The representation theory of the quasisimple groups is nearly identical to the projective representation theory of the simple groups.