A negative answer to the Kuroš–Černikov 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′≅Cp∞ 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)].