On a generalization of M-group

TL;DR

This paper generalizes Taketa's Theorem by showing that if every nonlinear irreducible character of a finite group is induced from a proper subgroup, then the group is solvable.

Contribution

It extends the class of groups characterized by induction properties of their irreducible characters, broadening understanding of group solvability conditions.

Findings

If each nonlinear irreducible character is induced from a proper subgroup, then the group is solvable.

Generalizes Taketa's Theorem on M-groups.

Provides a new criterion for group solvability based on character induction.

Abstract

In this paper, we will show that if for every nonlinear complex irreducible character of a finite group G, some multiple of it is induced from an irreducible character of some proper subgroup of G, then G is solvable. This is a generalization of Taketa's Theorem on the solvability of M-group.