¢. In Hartley’s

A negative answer to the Kuros) –C) ernikov Question 21 in [7], whether a group satisfying the normalizer condition is hypercentral, was given by Heineken and Mohamed in 1968 [6]. They constructed groups G satisfying: (i) G is a locally finite p-group for a prime p, (ii) G}G«FC p ¢ and G « is countable elementary abelian, (iii) every proper subgroup of G is subnormal and nilpotent, (iv) Z(G)¯1´, (v) the set of normal subgroups of G contained in G « is linearly ordered by set inclusion, see [3, p. 334], (vi) KG « is a proper subgroup in G for every proper subgroup K of G, see [6, Lemma 1(a)]. Similar constructions have been given in [10, 2, 3, 11]. In 1973 Hartley [3] gav