Research on M-groups of odd order

如果有限群G的一个线性特征由G的子群G的一个线性特征导出，则该群G的一个特征是单项式的。如果群G的所有复不可约特征都是单项式，则群G是m群。Isaacs问m群的Hall子群是否需要是m群。关于这个问题，我们给出了具有非m群的Hall子群的m群的例子。接下来，艾萨克提出了如下的猜想。具有不可约性质的可解群，作为两个忠实性质的乘积，必然是循环的。关于这个问题，我们证明了下面的定理。设G是一个m群，并假设G的某个不可约字符是两个忠实字符的乘积。那么G是循环的。

A character of a finite group G is monomial if it is induced from a linear character of a subgroup of G. A group G is an M-group if all its complex irreducible character are monomial. Isaacs asked whether Hall subgroups of M-group need be M-groups. Concerning this problem, we gave examples of M-groups that have Hall subgroups that are not M-groups.Next, Isaacs conjectured the following.Conjecture. A solvable group having an irreducible character that factors as a product of two faithful characters is necessarily cyclic.Concerning this problem, we proved the following theorem.Theorem. Let G be an M-group and suppose that some irreducible character of G is a product of two faithful characters. Then G is cyclic.