On the minimality property for locally finite groups

Shunkov, V. P.

doi:10.1007/bf02218982

Cited by 64 publications

(24 citation statements)

References 9 publications

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“…Consequently, G is locally finite. Therefore in view of Shunkov-KegelWehrfritz Theorem [14,15], G is Chernikov, which is a contradiction.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 93%

Some Minimal Conditions in Certain Extremely Large Classes of Groups

Chernikov¹

2015

J. Sib. Fed. Univ., Math. phys.

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Let L (respectively T) be the minimal local in the sense of D. Robinson class of groups, containing the class of weakly graded (respectively primitive graded) groups and closed with respect to forming subgroups and series. In the present paper, we completely describe: the L-groups with the minimal conditions for non-abelian subgroups and for non-abelian non-normal subgroups; the T-groups with the minimal conditions for (all) subgroups and for non-normal subgroups. By the way, we establish that every IHgroup, belonging to L, is solvable.

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“…Consequently, G is locally finite. Therefore in view of Shunkov-KegelWehrfritz Theorem [14,15], G is Chernikov, which is a contradiction.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 93%

Some Minimal Conditions in Certain Extremely Large Classes of Groups

Chernikov¹

2015

J. Sib. Fed. Univ., Math. phys.

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“…Recall that a group G is Chernikov if it has a subgroup of finite index that is a direct product of finitely many groups of type C p ∞ for various primes p (quasicyclic p-groups). By a deep result obtained independently by Shunkov [23] and Kegel and Wehrfritz [11] Chernikov groups are precisely the locally finite groups satisfying the minimal condition on subgroups, that is, any non-empty set of subgroups possesses a minimal subgroup.…”

Section: Resultsmentioning

confidence: 99%

On locally finite groups with a small centralizer of a four-subgroup

Romano

Shumyatsky

2011

Arch. Math.

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The main result of the paper is the following theorem. Let G be a locally finite group having a four-subgroup V such that CG(V ) is finite. Suppose that V contains two involutions v1 and v2 such that the centralizers CG(v1) and CG(v2) have finite exponent. Then G is almost locally soluble and [G, V ] has finite exponent. Since [G, V ] has finite index in G, the result gives a fairly detailed information about the structure of G.Mathematics Subject Classification (2010). 20F50, 20E25.

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“…Suppose now that G is periodic, and hence locally finite. If G is not aČernikov group, by a result ofŠunkov [11] it does not satisfy the minimal condition on abelian subgroups. Thus G contains an abelian subgroup A which is a direct product of infinitely many subgroups of prime order.…”

Section: Proofsmentioning

confidence: 99%