Groups which are factorized by subgroups of finite exponents. (English) Zbl 1089.20012

Let the group \(G=AB\) be the product of a locally soluble subgroup \(A\) of finite exponent and a locally finite subgroup \(B\) such that for each prime \(p\), \(A\) or \(B\) has no elements of order \(p\). It is shown that if \(G\) is an \(RN\)-group (for instance if it is locally soluble or radical), then \(G\) is locally finite-soluble. If an \(RN\)-group \(G\) is factorized by finitely many pairwise permutable locally finite subgroups \(A_i\) with finite mutually coprime exponents \(t_i\), where \(i=1,\dots,n\), then \(G\) is locally finite with finite exponent \(t=t_1t_2\cdots t_n\).
It should be noted that S. V. Ivanov has announced that every countable group can be embedded in a group which is the product of two groups which have all their subgroups of prime order \(p\). This implies that an arbitrary group which is the product of two subgroups of finite exponent, may have infinite exponent.


20E15 Chains and lattices of subgroups, subnormal subgroups
20F50 Periodic groups; locally finite groups
20E22 Extensions, wreath products, and other compositions of groups
Full Text: DOI


[1] Chernikov, N. S.: Groups which are Products of Permutable Subgroups, Nauk. Dumka, Kiev, 1987, 206 p. (in Russian). · Zbl 0699.20025
[2] Chernikov, N. S.: Groups which are factorized by pairwise permutable periodic subgroups without elements of the same prime ordes, In: Investigations of Groups with Restrictions for Subgroups, In-t Math. NAS Ukraine, Kiev, 1988, pp. 98-117 (in Russian). · Zbl 0743.20030
[3] Chernikov, N. S.: Factorization of linear groups and groups possessing a normal system with linear factors, Ukrain. Math. J. 40(3) (1988), 362-369.
[4] Chernikov, N. S.: The solavibility and the locally solvability of factorized RN-groups, In: Methods of Investigation of Algebraic and Topological Structures, Pedagogical Institute, Kiev, 1989, pp. 122-129 (in Russian).
[5] Chernikov, N. S.: Factorization of groups by periodic permutable subgroups without elements of the same prime orders, Ukrain. Math. J. 43(10) (1991), 1429-1436. · Zbl 0743.20031 · doi:10.1007/BF01061821
[6] Chernikov, N. S.: On groups factorizable by pairwise permutable almost locally normal subgroups, Ukrain. Math. J. 48(3) (1996), 429-431. · Zbl 0941.20028
[7] Chernikov, N. S.: Generalized soluble and generalized ?-soluble factorizable groups, Problems in Algebra (University Press, Gomel, Belarus) 10 (1996), 91-122 (in Russian). · Zbl 0912.20022
[8] Chernikov, N. S.: Periodic locally soluble groups factorizable by two locally nilpotent subgroups, Problems in Algebra (University Press, Gomel, Belarus) 11 (1997), 90-115 (in Russian). · Zbl 0912.20023
[9] Chernikov, S. N.: To the theory of locally solvable groups, Mat. Sb. 19(2-3) (1943), 317-333. · Zbl 0063.07318
[10] Chunikhin, S. A.: Subgroups of Finite Groups, Nauka i Tehnika, Minsk, 1964, 158 p. (in Russian). · Zbl 0119.02904
[11] Gross, F.: The 2-length of a finite solvable group, Pacific J. Math. 15(4) (1965), 1221-1237. · Zbl 0136.01501
[12] Hall, Ph. and Higman, J.: On the p-length of p-soluble groups and reduction theorems for Burnside?s problem, Proc. London Math. Soc. 6(21) (1956), 1-42. · Zbl 0073.25503 · doi:10.1112/plms/s3-6.1.1
[13] Ivanov, S. V.: On general products of some groups, In: XI All-Union Simposium on the Theory of Groups; Thesises of Reports, Kungurka, 31 January?2 February 1989, In-t of Math. and Mechanics of Ural Departament of Academy of Sciences of USSR, Sverdlovsk, 1989, pp. 147-148 (in Russian).
[14] Ivanov, S. V. and Ol?shanskii, A. Yu.: Some applications of graded diagrams in combinatorial group theory, In: London Math. Soc. Lecture Note Ser. 160, Cambridge Univ. Press, 1991, pp. 258-308.
[15] Kurosh, A. G.: The Theory of Groups, 3rd edn, Nauka, Moscow, 1967. · Zbl 0189.30801
[16] Ol?shanskii, A. Yu.: Groups of a bounded period with subgroups of a prime order, Algebra i Logika 21(5) (1982), 553-618.
[17] Plotkin, B. I.: Groups of Automorphisms of Algebraic Systems, Nauka, Moscow, 1966. · Zbl 0192.12302
[18] Shunkov, V. P.: On groups decomposed as an uniform product of their p-subgroups, Dokl. Akad. Nauk SSSR 154(3) (1964), 542-544.
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