A remark on \(M_ p\)-groups (Q1199254)

Let \(F\) be an algebraically closed field and let \(G\) be a finite field. An \(FG\)-module \(V\) is called monomial if \(V\) is induced from a 1- dimensional module. \(G\) is called an \(M\) (resp. \(M_ p\))-group if every simple \(FG\)-module is monomial where \(\text{char }F= 0\) (resp. \(\text{char }F = p\)). By Taketa's argument \(M\)- and \(M_ p\)-groups are solvable. Thus by Fong's liftability theorem \(M\)-groups are \(M_ p\)- groups for all primes \(p\). The converse is not true, for instance \(\text{SL}(2,3)\) is an \(M_ 2\)-group but not an \(M\)-group. In this note the authors prove that a \(p\)-nilpotent group \(G\) is an \(M\)-group if and only if \(G\) is an \(M_ p\)-group.