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

TL;DR

This paper proves that the transfinite E_α-envelopes of hypercentral subgroups are finitely solvable within groups satisfying chain conditions on centralizers, extending previous results on E_k envelopes.

Contribution

It introduces transfinite forms of E_α-envelopes and shows their finiteness and solvability for hypercentral subgroups, generalizing earlier work on E_k envelopes.

Findings

E_α-envelopes of hypercentral subgroups are solvable.

The descending chain of E_k envelopes terminates finitely.

Results extend previous findings on E_k envelopes.

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.