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)].