The Fong reduction revisited (Q1320207)

Let \(N\) be a normal subgroup of exponent \(n\) of the finite group \(G\), \(\zeta\) a \(G\)-invariant irreducible complex character of \(N\), \(F\) a field containing a primitive \(n\)th root of unity, and denote by \(Z\) the cyclic group of order \(n\). The main result of this paper is the following theorem, which is established using cohomological techniques developed by E. C. Dade and the author: \(\zeta\) and \(G\) determine a unique, up to equivalence, central extension \(Z\rightarrowtail G(\zeta) \twoheadrightarrow G/N\), if \(\widehat {G}\) is the fibre-product of \(G(\zeta)\) and \(G\) amalgamating \(G/N\), then there is an \(F\widehat {G}\)-module \(\widehat {V}\) whose restriction to \(N\) affords \(\zeta\) as a Brauer character; moreover, \(G(\zeta)\) is a split extension if and only if \(\widehat {V}\) can be chosen as an \(FG\)-module; finally, the exponent of \(G(\zeta)\) divides the exponent of \(G\). This theorem unifies and improves results which are known especially for algebraically closed fields, and also results of \textit{W. F. Reynolds} [J. Algebra 129, 481-493 (1990; Zbl 0691.20010)], and applies in the context of Fong reductions and the Fong-Swan theorem.