The partial Schur multiplier of a group

Refining the technique worked out in previous papers, we give a characterization, up to equivalence, of the factor sets sigma of partial projective representation of a group G over an algebraically closed field K in terms of equalities satisfied by sigma. This allows one to conclude that any component of The partial Schur multiplier pM(G) is an epimorphic image of a direct power of K*. Moreover, it is shown that any component of pM(G) is an epimorphic image of the component pM(GxG)(G) of the equivalence classes of the totally defined partial factor sets. Examples with cyclic G are considered, in particular, the total component pM(GxG)(G) is determined when G is an arbitrary finite cyclic group. In this case pM(GxG)(G) is a direct power of K*. (C) 2013 Elsevier Inc. All rights reserved.