The Upper Radical Property and Lower Radical Property of Groups

Abstract

We take in this paper an arbitrary class [Formula: see text] of groups as a base, and define a radical property 𝒫 for which every group in [Formula: see text] is 𝒫-semisimple. This is called the upper radical property determined by the class [Formula: see text]. At the same time, we define a radical property 𝒫 for which every group in [Formula: see text] is a 𝒫-radical group. This is called the first lower radical property determined by the class [Formula: see text]. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one.