[1] Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn(G) (its quotient by its center) is simple (and it follows Inn(G) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic).
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 representation theory of the quasisimple groups is nearly identical to the projective representation theory of the simple groups.
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