Groups where all the irreducible characters are super-monomial

TL;DR

This paper investigates super-monomial characters in groups, proving a conjecture for certain odd-order groups and providing examples of super M-groups with non-super M-group subgroups.

Contribution

It proves Isaacs's conjecture for odd-order M-groups where irreducible characters are prime-lifted, and constructs examples of super M-groups with non-super M-group subgroups.

Findings

Proved Isaacs's conjecture for specific odd-order M-groups.

Identified super M-groups with subgroups that are not super M-groups.

Provided explicit examples illustrating the structure of super M-groups.

Abstract

Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if $G$ is an $M$-group of odd order where every irreducible character is a $\{p \}$-lift for some prime $p$. We say that a group where irreducible character is super monomial is a super $M$-group. We use our results to find an example of a super $M$-group that has a subgroup that is not a super $M$-group.