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Barely Transitive and Heineken Mohamed Groups
Author(s) -
Belyaev V. V.,
Kuzucuoğlu M.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797004870
Subject(s) - mathematics , centralizer and normalizer , combinatorics , subgroup , normal subgroup , prime (order theory) , countable set , transitive relation , abelian group , group (periodic table) , lemma (botany) , nilpotent , pure mathematics , physics , ecology , poaceae , quantum mechanics , biology
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)].