×

zbMATH — the first resource for mathematics

On periodic groups of odd period \(n\geq 1003\). (English. Russian original) Zbl 1152.20034
Math. Notes 82, No. 4, 443-447 (2007); translation from Mat. Zametki 82, No. 4, 495-500 (2007).
Summary: Using the Adyan-Lysenok theorem claiming that, for any odd number \(n\geq 1003\), there is an infinite group each of whose proper subgroups is contained in a cyclic subgroup of order \(n\), it is proved that the set of groups with this property has the cardinality of the continuum (for a given \(n\)). Further, it is proved that, for \(m\geq k\geq 2\) and for any odd \(n\geq 1003\), the \(m\)-generated free \(n\)-periodic group is residually both a group of the above type and a \(k\)-generated free \(n\)-periodic group, and it does not satisfy the ascending and descending chain conditions for normal subgroups either.

MSC:
20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
20E28 Maximal subgroups
20E07 Subgroup theorems; subgroup growth
20E32 Simple groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Higman, ”A finitely generated infinite simple group,” J. London Math. Soc. 26, 61–64 (1951). · Zbl 0042.02201 · doi:10.1112/jlms/s1-26.1.61
[2] S. I. Adyan, ”Periodic products of groups,” in Trudy Mat. Inst. Steklov. Vol. 142: Number Theory, Mathematical Analysis and Their Applications (Nauka, Moscow, 1976), pp. 3–21 [Proc. Steklov Inst. Math., Vol. 142, pp. 1–19 (1979)].
[3] S. I. Adyan, ”The simplicity of periodic products of groups,” Dokl. Akad. Nauk SSSR 241(4), 745–748 (1978) [Soviet Math. Dokl. 19 (4), 910–913 (1978) (1979)]. · Zbl 0416.20024
[4] H. Neumann, Varieties of Groups (Springer-Verlag, New York, 1967; Mir, Moscow, 1969). · Zbl 0149.26704
[5] A. Yu. Ol’shanskii, ”Groups of bounded period with subgroups of prime order,” Algebra i Logika 21(5), 553–618 (1982) [Algebra and Logic 21 (5), 369–418 (1982)].
[6] V. S. Atabekyan and S. V. Ivanov, Two Remarks on Groups of Bounded Period, Available from VINITI, No. 243-V87 [in Russian].
[7] S. I. Adyan and I. G. Lysenok, ”Groups all of whose proper subgroups are finite cyclic,” Izv. Akad. Nauk SSSR Ser. Mat. 55(5), 933–990 (1991).
[8] V. S. Atabekyan, On the Approximation and Subgroups of Free Periodic Groups, Available from VINITI, No. 5380-V86 [in Russian].
[9] A. Yu. Ol’shanskii, Geometry of Defining Relations in Groups (Nauka, Moscow, 1989; Kluwer Academic Publishers Group, Dordrecht, 1991).
[10] S. I. Adyan, ”Normal subgroups of free periodic groups,” Izv. Akad. Nauk SSSR Ser. Mat. 45(5), 931–947 (1981) [Math. USSR-Izv. 19(2), 215–229 (1982)].
[11] S. I. Adyan, The Burnside Problem and Identities in Groups (Nauka, Moscow, 1975; Springer-Verlag, Berlin-New York, 1979). · Zbl 0306.20045
[12] N. Gupta and F. Levin, ”Generation groups of certain product varieties,” Arch. Math. 30(2), 113–117 (1978). · Zbl 0406.20024 · doi:10.1007/BF01226028
[13] V. L. Shirvanyan [Ĺ irvanjan], ”Embedding the group B(, n) in the group B(2, n),” Izv. Akad. Nauk SSSR Ser. Mat. 40(1), 190–208 (1976) [Math. USSR-Izv. 10 (1976), 181–199 (1977)]. · Zbl 0336.20027
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.