Characters of maximal subgroups of \(M\)-groups
An \(M\)-group is a finite group all of whose complex irreducible characters are monomial. It is a conjecture, still unsolved, that any normal subgroup of an \(M\)-group of odd order, is itself an \(M\)-group. For \(M\)-groups of even order the corresponding conjecture is false [see the reviewer, Indag. Math. 36, 157-167 (1974; Zbl 0279.20013); and \textit{E. C. Dade}, Math. Z. 133, 313-317 (1973; Zbl 0255.20007)]. Therefore a suitable oddness hypothesis attracted the attention of several character-theorists, during a long period of time, when dealing with \(M\)-group character theory. The author's (long) paper is a very attractive contribution to the (already vast) amount of theory on \(M\)-groups. It contains intrinsic results for other research in character theory as well. In respect to \(M\)-group results obtained here, we mention only: Theorem A. Let \(G\) be an \(M\)-group, let \(H\) be a maximal subgroup of \(G\) such that \(|G:H|\) is odd. Then \(\theta(1)^2\) divides \(|G:H|\) for each primitive \(\theta\in\text{Irr}(H)\).
scientific article; zbMATH DE number 923707