On subgroups normalized by the base group in monomial groups, and their centralizers

Abstract

A large family of examples for subgroups normalized by the base group in a complete monomial group of finite degree over any group $H$ is given. This is then proven to be a complete characterization of such subgroups in the case of an abelian group $H$. Centralizer structure for this kind of subgroups, even in the non abelian case, is completely determined. Notably, separate study of the case of elementary abelian $2$-group $H$ is needed. In the last part, the results are extended to the case of limit monomial groups.