Factorizations of locally finite groups. (English. Russian original) Zbl 0598.20034

Sib. Math. J. 21, 890-897 (1981); translation from Sib. Mat. Zh. 21, No. 6, 186-195 (1980).
Main results: Theorem 1. Let the group G be locally finite and factorizable by locally normal subgroups A and B, \(\pi\) a set of primes, P and Q invariant Sylow \(\pi\)-subgroups of A and B, respectively. Then if G is locally solvable or the set \(\pi\) consists of a single prime, P and Q commute with one another and the product \(P\cdot Q\) is a Sylow \(\pi\)- subgroup of G. Furthermore, every finite \(\pi\)-subgroup of G is contained in some subgroup conjugate in G to \(P\cdot Q\). Theorem 2. Let G be a locally finite group which is factorized by two almost locally normal subgroups which satisfy the minimal condition for p-subgroups (p a prime). Then G also satisfies the minimal condition for p-subgroups.


20E15 Chains and lattices of subgroups, subnormal subgroups
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20F50 Periodic groups; locally finite groups
Full Text: DOI


[1] B. Huppert, Endliche Gruppen, Vol. 1, Springer-Verlag, New York (1967).
[2] H. Heineken, ?Maximale p-Untergruppen lokal endlicher Gruppen?, Arch. Math.,23, No. 4, 351-361 (1972). · Zbl 0246.20021
[3] Kourovskii Notebook [in Russian], Novosibirsk, Izd. Inst. Mat. Sib. Otd. Akad. Nauk SSSR (1967).
[4] M. I. Kargapolov and Yu. I. Merzlyakov, Foundations of Group, Theory [in Russian], Nauka, Moscow (1977). · Zbl 0499.20001
[5] B. Amberg, ?Artinian and noetherian factorized groups?, Rend. Sem. Mat. Univ. Padova,55, 105-122 (1976). · Zbl 0375.20021
[6] N. Ito, ?Über das Produkt von zwei abelschen Gruppen? Math. Z.,62, No. 4, 400-401 (1955). · Zbl 0064.25203
[7] A. G. Kurosh, Group Theory, Chelsea Publ.
[8] M. Hall, Theory of Groups, Macmillan (1961). · Zbl 0104.02201
[9] V. S. Charin, ?On the minimality condition for normal subgroups of locally solvable groups?, Mat. Sb.,33, No. 1, 27-36 (1953).
[10] V. P. Shunkov, ?On locally finite groups with the minimality condition for Abelian subgroups?, Algebra Logika,9, No. 5, 579-615 (1970).
[11] T. A. Neeshpapa and N. F. Sesekin, ?On some criteria for extremality of locally nilpotent groups?, in: Algebra and Mathematical Analysis [in Russian], Proc. Sverdlovsk State Pedagog. Inst., Vol. 219 (1974), pp. 55-62. · Zbl 0328.20026
[12] Ya. D. Polovitskii, ?Locally finite groups in which almost all elements have finite centralizers?, in: Group Theory and Some Problems in Algebra [in Russian], Scientific Proc. Perm’ State Univ., Vol. 343 (1975), pp. 56-62.
[13] N. N. Myagkova, ?On groups of finite rank?, Izv. Akad. Nauk SSSR, Ser. Mat.,13, No. 6, 495-512 (1949).
[14] M. I. Kargapolov, ?Locally finite groups having normal systems with finite quotients?, Sib. Mat. Zh.,2, No. 6, 853-873 (1962).
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.