On The $E_\alpha$-Envelopes of Hypercentral Subgroups

Abstract

The E_{k} envelopes that generalize the double centralizers form a descending chain. In this paper we show that this descending chain stops after finitely many steps for hypercentral subgroups by defining the transfinite forms of some basic descriptions. In particular, we prove that the E_{{\alpha}}- envelopes of hypercentral subgroups are solvable in the class of groups satisfying chain condition on centralizers. These extend previous results on E_{k} envelopes.