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 σ of partial projective representation of a group G over an algebraically closed field K in terms of equalities satisfied by σ. 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 pMG×G(G) of the equivalence classes of the totally defined partial factor sets. Examples with cyclic G are considered, in particular, the total component pMG×G(G) is determined when G is an arbitrary finite cyclic group. In this case pMG×G(G) is a direct power of K⁎.