Universal perfect central extension
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H2(G, Z) of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.
The Schur multiplier M(G) of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of M(G) is not divisible by p. In particular, if all Sylow p-subgroups of G are cyclic, then M(G) is trivial.
For instance, the Schur multiplier of the nonabelian group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2.
The Schur multipliers of the finite simple groups are given at the list of finite simple groups. The covering groups of the alternating and symmetric groups are of considerable recent interest.
Schur's original motivation for studying the multiplier was to classify projective representations of a group, and the modern formulation of his definition is the second cohomology group H2(G, C×). A projective representation is much like a group representation except that instead of a homomorphism into the general linear group GL(n, C), one takes a homomorphism into the projective general linear group PGL(n, C). In other words, a projective representation is a representation modulo the center.
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