Let G be a finite group. Let 1 - I, is defined by Z,+,(G)/Z,(G) = Z(G/Z,(G)). Let H(G) == ui ZJG). The subgroup H(G) is called the hypercenter of G. Clearly H(G) is nilpotent and characteristic in G. The purpose of this paper is to prove some necessary and sufficient conditions for an element or a subgroup of a finite group G to lie in H(G). Baer [l] studied the properties of H(G) and obtained a number of characterizations of H(G). 0 ne of Baer’s results is the following: a p-element g of a finite group G belongs to H(G) if and only if g commutes with every p’-element of G. Now suppose that we do not require g to commute with all the $-elements of G but only with all the p’-elements of every soluble subgroup of G which contains g. We show that except when p = 2 this is sufficient to ensure that g belongs to H(G). A result of this kind which involves only a “local” hypothesis was proved by Shult [7]. He showed that an Abelian subgroup R of a finite group G lies in Z(G) f i and only if =1 lies in the center of every soluble subgroup of G which contains d. Let G be a finite group. Denote by E(G) the set of all subgroups of G which have order divisible by at most two distinct primes. For any subgroup A of prime power order let eA(G) be the subset of elements of g(G) which contain A. It follows from the celebrated pa@-theorem of Burnside that all the subgroups in V(G) are soluble. (See, for example, [5, p. 4921.) The main result of this paper is the following proposition.