On locally finite groups of finite rank

Shunkov, V. P.

doi:10.1007/bf02219979

Cited by 55 publications

(24 citation statements)

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“…Thus R has finite rank and so, by [ It remains only to deal with the case where G is locally finite. By [14] G has an abelian subgroup A of infinite rank, and since A in turn has an infinite descending chain of subgroups of infinite rank it follows easily that there is a subgroup C of A such that C g is a proper subgroup of C for some g in G. But g has finite order n, say, and since C g i < C g i−1 for each positive integer i we obtain the contradiction that C = C g n < C. This completes the proof.…”

Section: Lemma 4 Let X Be a Finitely Generated Nilpotent-by-finite Gmentioning

confidence: 99%

“…As usual, γ i (X) denotes the ith term of the lower central series of a group X, and the Fitting radical Fitt(X) of X is the product of all normal nilpotent subgroups of X. (c) A locally finite group of finite rank is almost locally soluble, that is, it has a normal locally soluble subgroup of finite index [14]. Indeed, the same holds for locally (soluble-by-finite) groups, by [3] (see also [4]).…”

Section: Is Countable Then H and K May Be Chosen So That I G (H) = G mentioning

confidence: 99%

“…For certain classes of groups G this suffices to ensure that G itself has finite rank. For example, this is the case for locally finite groups [14], locally nilpotent groups (see [10,Theorem 6.36 and its corollaries]), and for hyperabelian (and even radical) groups [1]. It is not the case for locally soluble groups, as shown in [9].…”

Section: Introductionmentioning

confidence: 99%

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Locally (Soluble-by-Finite) Groups With Various Restrictions on Subgroups of Infinite Rank

2005

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Section: Lemma 4 Let X Be a Finitely Generated Nilpotent-by-finite Gmentioning

confidence: 99%

Section: Is Countable Then H and K May Be Chosen So That I G (H) = G mentioning

confidence: 99%

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Locally (Soluble-by-Finite) Groups With Various Restrictions on Subgroups of Infinite Rank

2005

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“…In particular, if H is soluble then so is Kin other words, each insoluble section K of G has no soluble subgroups of infinite rank. Now a locally nilpotent group of infinite rank has an infinite rank abelian subgroup [7, Corollary 2 to Theorem 6.36], and the same holds for hyperabelian groups [1] and locally finite groups [12]. Again let K be an infinite rank section of G. If M is an infinite rank subgroup of K generated by K-invariant subgroups M λ of finite rank then, since each M λ has a characteristic and hence K-invariant abelian series [7,Lemma 10.39], M is hyperabelian and K is therefore soluble (since M has an abelian subgroup of infinite rank).…”

Section: The Proof Of Theoremmentioning

confidence: 99%

Groups in Which All Subgroups of Infinite Rank Are Subnormal

Kurdachenko

Smith²

2004

Glasgow Math. J.

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Abstract. Let G be a locally soluble-by-finite group in which every nonsubnormal subgroup has finite rank. It is proved that either G has finite rank or G is soluble and locally nilpotent (and even a Baer group). On the other hand, a group G is constructed that has infinite rank and satisfies the given hypothesis, but does not have every subgroup subnormal.2000 Mathematics Subject Classification. 20E15, 20F19.

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“…We may now apply a deep theorem of Sunkov (1971) to conclude that K has a locally soluble normal subgroup K o of finite index and finite rank. By Kargapolov (1959) …”

Section: B Let G Be a Locally Finite Group And Let N = Tt(g) Then Nmentioning

confidence: 99%

The Schur-Zassenhaus theorem in locally finite groups

Hartley

1976

J. Aust. Math. Soc.

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Let L = HK be a semidirect product of a normal locally finite n '-group H by a locally finite 7r-group K, where n is a set of primes. Suppose C K (H) = 1 and L is Sylow w-sparse (which in the countable case just says that the Sylow 7r-subgroups of L are conjugate). This paper completes the characterization of those groups which can occur as K-this had previously been obtained under the assumption that L is locally soluble. The answer is the same-essentially that the groups occurring are those having a subgroup of finite index which is a subdirect product of so-called "pinched" groups.This short note is an addendum to my paper Hartley (1975), with which familiarity will be assumed and from which any unexplained terminology is drawn. Its purpose is to point out that certain hypotheses of local solubility in one of the main results of that paper are superfluous. We improve our previous Theorem B to THEOREM B. Let G be a locally finite group and let n = TT(G). Then necessary and sufficient conditions that there exist a locally finite Sylow Tr-sparse group L = HK such that H

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On locally finite groups of finite rank

Cited by 55 publications

References 9 publications

Locally (Soluble-by-Finite) Groups With Various Restrictions on Subgroups of Infinite Rank

Locally (Soluble-by-Finite) Groups With Various Restrictions on Subgroups of Infinite Rank

Groups in Which All Subgroups of Infinite Rank Are Subnormal

The Schur-Zassenhaus theorem in locally finite groups

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