Connecting monomiality questions with the structure of rational group algebras

TL;DR

This paper connects the study of monomial groups with the structure of rational group algebras, proving embedding theorems and exploring new classes of generalized strongly monomial groups to unify two research directions.

Contribution

It proves that finite solvable groups embed into generalized strongly monomial groups, linking monomial group theory with rational group algebra structure.

Findings

Embedding of solvable groups into generalized strongly monomial groups

Correlation of monomiality questions with algebraic structures

Raised new questions weaker than classical monomiality questions

Abstract

In recent times, there has been a lot of active research on monomial groups in two different directions. While group theorists are interested in the study of their normal subgroups and Hall subgroups, the interest of group ring theorists lie in the structure of their rational group algebras due to varied applications. The purpose of this paper is to bind the two threads together. Revisiting Dade's celebrated embedding theorem which states that a finite solvable group can be embedded inside some monomial group, it is proved here that the embedding is indeed done inside some generalized strongly monomial group. The so called generalized strongly monomial groups arose in a recent work of authors while understanding the algebraic structure of rational group algebras. Still unresolved monomiality questions have been correlated by proving that all the classes of monomial groups where they have been answered are generalized strongly monomial. The study also raises some intriguing questions weaker than those asked by Dornhoff and Isaacs in their investigations.