PROPERTIES OF SUBGROUPS OF SOLVABLE GROUPS THAT IMPLY THEY ARE NORMALLY EMBEDDED

Abstract

Recently, Ballester-Bolinches [1 and 2], Pedraza-Aguilera [2] and Perez-Ramos [2] have studied circumstances under which certain injectors and projectors, which are always pronormal, must be normally embedded. In this note we give a scheme for describing a minimal counterexample to a conjecture of the form: a subnormally embedded subgroup with properties $\alpha_1$, $\alpha_2,{\ldots\,},\alpha_{n}$ is normally embedded, where $\alpha_1$, $\alpha_2,{\ldots\,},\alpha_{n}$ satisfy certain conditions. We then show contradictions in certain cases involving finite solvable groups.