The Schur multiplier, fields, roots of unity, and a natural splitting (Q582381)

Let G be a finite group and let \(\Omega\) be a subgroup of \({\mathbb{C}}^*\) which contains a root of unity of order \(| G|\). The universal coefficient theorem yields a split exact sequence \[ 1\quad \to \quad Ext(G/G',\Omega)\quad \to \quad H^ 2(G,\Omega)\quad \to \quad H^ 2(G,{\mathbb{C}}^*)\quad \to \quad 1 \] where the splitting is known to be natural in \(\Omega\). One result of this paper provides a natural splitting in G as well as \(\Omega\). If \(\Omega =K^*\) for a field K the author points out a connection with the problem of realizing complex projective representations of G and its subgroups over the field K. He considers also the case where the field \({\mathbb{C}}\) is replaced by any algebraically closed field. The results of this paper might also be useful in those parts of number theory where the group \(H^ 2(G,{\mathbb{C}}^*)\) is used.