Barely Transitive and Heineken Mohamed Groups

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 ′≅ C 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)].