Irreducible products of characters.
Let \(G\) be a finite group, \(\alpha,\beta\in\text{Irr}(G)\). In this paper the author asks when it can happen that the product \(\alpha\beta\in\text{Irr}(G)\). He considers the most interesting case where \(\alpha\) and \(\beta\) are faithful. ``We have been unable to find a noncyclic solvable example (with irreducible \(\alpha\beta\)), however, and we suspect that none exists. We can prove, at least, that if such example does exist, it would have a fairly complicated structure.'' Theorem A + B. Suppose that \(G\) is solvable and that some irreducible character of \(G\) factors as a product of two faithful characters \(\alpha\), \(\beta\). If \(G\) is not cyclic, then its Fitting height is at least 4. Next, \(\alpha(1)\) and \(\beta(1)\) have at least two common prime factors. Also, \(G\) has a noncentral minimal normal subgroup, and if \(M\) is any such subgroup, say \(p\)-subgroup, then \(p\) divides \(\alpha(1)\), \(\beta(1)\), and the rank of \(M\). Furthermore, the group of automorphisms of \(M\) induced by \(G\) is not \(p\)-nilpotent. The following module-theoretic conjecture is connected closely with the problem considered. Conjecture C. Let \(V\) be an irreducible \(FG\)-module, where \(F\) is a field. Suppose that we can factor \(G=XY\), where \(X\) and \(Y\) are subgroups of \(G\), and each of \(X\) and \(Y\) has a nonzero fixed point in \(V\). Then \(G\) acts trivially on \(V\). Theorem D. Assume that Conjecture C holds for some solvable group \(G\). If \(G\) has an irreducible character \(\alpha\beta\) with faithful characters \(\alpha\) and \(\beta\), then \(G\) is cyclic. Some intermediate results are proved. For example, Lemma (Lemma 2.1 + Corollary 2.2): Let \(\alpha,\beta\in\text{Irr}(G)\) and \(\alpha\beta\) is irreducible. (a) If \(\alpha\) is induced from some subgroup \(X\leq G\), then \(\beta_X\) is irreducible. Also, if \(\beta\) is induced from \(Y\leq G\), then \(G=XY\). (b) Every cyclic normal subgroup of \(G\) is central.
scientific article; zbMATH DE number 1423891