On minimal non \(M_ p\)-groups (Q689692)

We study the structure of minimal non \(M_ p\)-groups. A finite group \(G\) is called an \(M_ p\)-group if every irreducible \(FG\)-module is monomial, where \(F\) is an algebraically closed field of characteristic \(p>2\). This is a \(p\)-modular analogue of the definition of \(M\)-groups. The structure of minimal non \(M\)-groups was studied by \textit{D. T. Price} [Math. Z. 130, 325-337 (1973; Zbl 0241.20007)], and using it, we obtain the similar result for minimal non \(M_ p\)-groups. Also we prove that some minimal non \(M_ p\)-groups cannot be normal subgroups of \(M_ p\)-groups.