# Schur multiplicator

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Schur multiplier, of a group The cohomology group , where is the multiplicative group of complex numbers with trivial -action. The Schur multiplicator was introduced by I. Schur  in his work on finite-dimensional complex projective representations of a group (cf. Projective representation). If is such a representation, then can be interpreted as a mapping such that where is a -cocycle with values in . In particular, the projective representation is the projectivization of a linear representation if and only if the cocycle determines the trivial element of the group . If , then is called a closed group in the sense of Schur. If is a finite group, then there exist natural isomorphisms Let . If a central extension (*)

of a finite group is given, then there is a natural mapping whose image coincides with . This mapping coincides with the mapping induced by the cup-product with the element of defined by the extension (*). Conversely, for any subgroup there is an extension (*) such that . If , then the extension (*) is uniquely determined by the homomorphism . If is a monomorphism, then any projective representation of is induced by some linear representation of .

How to Cite This Entry:
Schur multiplicator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_multiplicator&oldid=12190
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article