Theory of minimax groups

Zaitsev, D. I.

doi:10.1007/bf01091652

Cited by 15 publications

(19 citation statements)

References 7 publications

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“…Let L be the third term of the lower central series of G. Assume that L is not soluble-by-finite. By the above-quoted result of Za~tsev [14], there exists a chain of subgroups of L:…”

Section: A##eamentioning

confidence: 91%

Groups with a maximality condition for some non-normal subgroups

Cutolo

Kurdachenko²

1995

Geom Dedicata

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“…Let L be the third term of the lower central series of G. Assume that L is not soluble-by-finite. By the above-quoted result of Za~tsev [14], there exists a chain of subgroups of L:…”

Section: A##eamentioning

confidence: 91%

Groups with a maximality condition for some non-normal subgroups

Cutolo

Kurdachenko²

1995

Geom Dedicata

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“…According to [14], in G 1 and G 2, there exist normal divisible Abelian Chernikov subgroups D l and D 2 such that the quotient groups G 1 /D l and G 2 / D 2 contain subgroups of finite index with finite rational series. This implies that D 1 < D 2 and, hence, the union of divisible parts of all finitely generated subgroups of the group G is its periodic Abelian normal subgroup.…”

Section: (3)mentioning

confidence: 99%

Groups of finite non-Abelian sectional rank

Dashkova¹

1997

Ukr Math J

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We study non-Abelian locally finite groups and non-Abelian locally solvable groups of finite non-Abelian sectional rank and prove that their (special) rank is finite.The notion of non-Abelian sectional rank of a group was introduced in [1]. In what follows, a section of a group G is understood as a quotient group A/B, where A and B are nonidentity subgroups of the group G, and the subgroup B is normal in A. Recall that a non-Abelian sectional rank of a non-Abelian group G is the minimum number r for which any non-Abelian finitely generated section of the group G can be generated by at most r elements. If all sections of the non-Abelian group G are Abelian, the non-Abelian sectional rank of the group G is assumed to be equal to zero. If G has at least one non-Abelian section and the number r with the indicated properties does not exist, then the non-Abelian sectional rank of the group G is regarded as infinite. We use the notation ~c(G) for the non-Abelian sectional rank of the group G. As is customary, by symbols r(G) and r 0 (G) we denote, respectively, the special rank and the 0-rank of the group G.The finiteness of the non-Abelian sectional rank of a group results in the finiteness of its (special) rank in the classes of non-Abelian locally nilpotent groups [1] and non-Abelian solvable groups [2]. The aim of the present work is to prove a similar assertion for a non-Abelian locally finite group and a non-Abelian locally solvable group. Theorem 1. A non-Abelian locally finite group of finite non-Abelian sectional rank has a finite (special) rank.

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“…The weak minimal condition for non-P subgroups is defined similarly. Zaičev proved in [16] that a locally (soluble-by-finite) group G satisfying either the weak maximal or the weak minimal condition for all subgroups is a soluble-by-finite minimax group, that is, G has a normal soluble subgroup H of finite index which in turn has a finite normal series whose factors are abelian and satisfy either max or min. In particular, G has finite rank.…”

Section: Theorem 1 Let G Be a Locally (Soluble-by-finite) Group (I)mentioning

confidence: 99%

“…Again we may assume that G is countable and locally soluble, also that G has no nontrivial normal subgroups of finite rank. If N is a nontrivial normal subgroup of G then N has infinite rank and so G/N satisfies the weak maximal condition for all subgroups and is therefore soluble minimax [16], so the finite residual of G/N is radicable (or "divisible") nilpotent, by [10, Theorem 9.31]. The intersection V of all nontrivial normal subgroups N of G is trivial, for if not then V is a chief factor of G and hence abelian and therefore of finite rank, a contradiction.…”

Section: Lemma 2 Let G Be a Group That Is The Ascending Union Of Submentioning

confidence: 99%