  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.
