  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.
