Groups - Korea 94
DOI: 10.1515/9783110908978.189
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On Locally Graded Groups

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Cited by 19 publications
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“…Then by ([10] 31.12), there is a finite normal series with cyclic factors H = N 0 N 1 ... N m = N. We know that N 0 is finitely generated and assume, that N i is finitely generated. Since by assumption G does not contain free non-cyclic subsemigroups, and the group N i /N i+1 is cyclic, it follows from ( [5], Lemmas 5 and 1) that N i+1 is finitely generated, which accomplishes the induction, and proves that N is finitely generated. 2 Lemma 2 A finitely generated group, which is finite-by-nilpotent-by-finite, is nilpotent-by-finite.…”
mentioning
confidence: 76%
“…Then by ([10] 31.12), there is a finite normal series with cyclic factors H = N 0 N 1 ... N m = N. We know that N 0 is finitely generated and assume, that N i is finitely generated. Since by assumption G does not contain free non-cyclic subsemigroups, and the group N i /N i+1 is cyclic, it follows from ( [5], Lemmas 5 and 1) that N i+1 is finitely generated, which accomplishes the induction, and proves that N is finitely generated. 2 Lemma 2 A finitely generated group, which is finite-by-nilpotent-by-finite, is nilpotent-by-finite.…”
mentioning
confidence: 76%
“…It is shown by Kim and Rhemtulla [11], that every locally graded n-Engel group G is locally nilpotent. Then by a result of Burns and Medvedev [6], G is contained in the variety N c(n) B e(n) ∩ B e(n) N c(n) , where c(n) and e(n) depend on n only.…”
Section: Theorem 4 An N-engel Group G Satisfies a Positive Law If G Imentioning
confidence: 99%
“…This gives a negative answer to Problem 1 in general, however we show in Theorem 2, that in the class of locally graded groups the answer is affirmative. Proof It is shown by Kim and Rhemtulla [5], that every locally graded nEngel group G is locally nilpotent. Then by a result of Burns and Medvedev [10], G is contained in a variety N c B e ∩ B e N c , where c and e depend on n only.…”
Section: Problemsmentioning
confidence: 99%
“…Let H be a two-generator subgroup in G. Since H also is locally graded and collapsing, then by ( [5] Theorem A and Lemma 3), H contains a polycyclic normal subgroup N of finite index. Then by the result of Rosenblatt ([11], 4.12), the finitely generated soluble F-group N must be nilpotent-by-finite.…”
Section: Problemsmentioning
confidence: 99%