# On super-monomial characters and groups having two irreducible monomial character degrees

## TL;DR

This paper investigates super-monomial characters in odd M-groups, proving that lowest degree non-linear irreducible characters are super-monomial and exploring conditions for super-monomiality in normal subgroups, along with groups having two monomial degrees.

## Contribution

It proves that all lowest degree non-linear irreducible characters in odd M-groups are super-monomial and identifies conditions for super-monomiality in certain subgroups.

## Key findings

- Lowest degree non-linear irreducible characters are super-monomial in odd M-groups.
- Conditions are provided for super-monomiality in normal subgroups.
- Analysis of groups with exactly two irreducible monomial character degrees.

## Abstract

A character of a group is said to be super-monomial if every primitive character inducing it is linear. It is conjectured by Isaacs that every irreducible character of an odd $M$-group is super-monomial. We show that all non linear irreducible characters of lowest degree of an odd $M$-group is super-monomial and provide cases in which one can guarantee that certain irreducible characters of normal subgroups are super-monomial. Finally, we study groups having two irreducible monomial character degrees.