×

zbMATH — the first resource for mathematics

Residually finite groups with nontrivial intersections of pairs of subgroups. (English. Russian original) Zbl 0956.20018
Sib. Math. J. 41, No. 2, 362-365 (2000); translation from Sib. Mat. Zh. 41, No. 2, 437-441 (2000).
The author proves that every nonabelian residually finite group in which every pair of subgroups has nontrivial intersection is a finite generalized quaternion group.

MSC:
20E26 Residual properties and generalizations; residually finite groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20E34 General structure theorems for groups
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] Shunkov V. P., ”On a certain class ofp-groups,” Algebra i Logika,4, No. 4, 484–496 (1970).
[2] Adyan S. I., ”On certain torsion-free groups,” Izv. Akad. Nauk SSSR Ser. Mat.,35, No. 3, 459–468 (1971). · Zbl 0259.20027
[3] Ol’shanskiî, A. Yu., ”A remark on a countable topological group,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 3, 103 (1980).
[4] Sozutov A. I., On the Existence of Infinite Subgroups with Nontrivial Locally Finite Radical in a Group [in Russian] [Preprint], Krasnoyarsk. Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Krasnoyarsk, 1980, pp. 11–19.
[5] Sozutov A. I., ”An example of an infinite finitely generated Frobenius group,” in: Abstracts: 7th All-Union Symposium on Group Theory [in Russian], Krasnoyarsk, 1980, p. 116.
[6] Shirvanyan V. L., ”Noncommutative periodic groups with nontrivial intersection of all cyclic subroups,” in: Abstracts: 7th All-Union Symposium on Group Theory [in Russian], Krasnoyarsk, 1980, p. 137.
[7] Ol’shanskiî A. Yu., The Geometry of Defining Relations in Groups [in Russian], Nauka, Moscow (1989).
[8] Sozutov A. I., ”On the structure of the noninvariant factor in some Frobenius groups,” Sibirsk. Mat. Zh.,35, No. 4, 893–901 (1994). · Zbl 0908.70013
[9] Kurosh A. G., The Theory of Groups [in Russian], Nauka, Moscow (1967). · Zbl 0189.30801
[10] Obraztsov V. N., ”Infinite periodic residually finite groups with prescribed properties,” in: Kurosh Algebraic Conference’98, Abstracts of Talks, Moscow, 1998, pp. 94–95.
[11] Hall M., The Theory of Groups [Russian translation], Izdat. Inostr. Lit., Moscow (1962). · Zbl 0115.23001
[12] Kargapolov M. I. andMerzlyakov Yu. I., Fundamentals of the Theory of Groups [in Russian], Nauka, Moscow (1977).
[13] Fuchs L., Infinite Abelian Groups. Vol. 2 [Russian translation], Mir, Moscow (1977). · Zbl 0366.20037
[14] Gorchakov Yu. M., Groups with Finite Classes of Conjugate Elements [in Russian], Nauka, Moscow (1978). · Zbl 0496.20025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.