Schur Multiplier - Relation To Projective Representations

Relation To Projective Representations

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.

Schur (1904, 1907) showed that every finite group G has associated to it at least one finite group C, called a Schur cover, with the property that every projective representation of G can be lifted to an ordinary representation of C. The Schur cover is also known as a covering group or Darstellungsgruppe. The Schur covers of the finite simple groups are known, and each is an example of a quasisimple group. The Schur cover of a perfect group is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to isoclinism.