Variants of theorems of Baer and Hall on finite-by-hypercentral groups

We show that if a group G has a finite normal subgroup L such that G/L is hypercentral, then the index of the (upper) hypercenter of G is at most |Aut(L)|⋅|L|. It follows an explicit bound for |G/Z2m(G)| in terms of d=|γm+1(G)| and independent of m∈N, provided d is finite. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we extend other results in the literature by applying our results to groups of automorphisms acting in a restricted way on an ascending normal series of a group G.