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Journal of the London Mathematical Society 1997 55(2):261-263; doi:10.1112/S0024610797004870
© 1997 by London Mathematical Society
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© The London Mathematical Society

Barely Transitive and Heineken Mohamed Groups

V. V. Belyaev and M. Kuzucuoglu

Krasnoyarsk Engineering Building Institute (KISI) Krasnoyarsk 660041, Russia. E-mail: belyaev{at}home.krasnoyarsk.su
Department of Mathematics, Middle East Technical University 06531 Ankara, Turkey. E-mail: matmah{at}rorqual.cc.metu.edu.tr

Received 24 September 1996. Revision received 3 January 1997.

A negative answer to the Kuros–Cernikov 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'{cong}Cp{infty} 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)].


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