Abstract
In the paper, the structure of Shunkov groups and periodic groups with a finite non-Engel element that are saturated with finite Frobenius groups and contain a nontrivial locally finite radical is determined.
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References
A. K. Shlepkin, “On certain torsion groups saturated with finite simple groups,” Siberian Adv. Math. 9 (2), 100–108 (1999).
A. I. Sozutov and V. P. Shunkov, “On a generalization of Frobenius’ theorem to infinite groups,” Math. USSR-Sb. 29 (4), 441–451 (1976).
A. G. Kurosh, The Theory of Groups (Chelsea Publishing Co., New York, NY, 1960).
A. K. Shlepkin, “Conjugately biprimitively finite groups containing finite unsolvable subgroups,” in Third International Conference in Algebra, August 23–28, 1993, Collection of Abstracts (Krasnoyarsk, 1993), p. 369 [in Russian].
A. I. Sozutov and V. P. Shunkov, “Infinite groups saturated by Frobenius subgroups,” Algebra Logic 16 (6), 477–493 (1978).
A. I. Sozutov and V. P. Shunkov, “Infinite groups saturated by Frobenius subgroups. II,” Algebra Logic 18 (2), 129–139 (1979).
A. K. Shlepkin, “Conjugately biprimitively finite groups saturated by finite simple subgroups \(U_3(2^n)\),” Algebra Logic 37 (5), 345–350 (1998).
A. K. Shlepkin, “On the periodic part of some Shunkov groups,” Algebra Logic 38 (1), 51–66 (1999).
A. I. Sozutov and A. K. Shlepkin, “On some groups with finite involution saturated with finite simple groups,” Math. Notes 72 (3), 398–410 (2002).
D. V. Lytkina and V. D. Mazurov, “Periodic groups saturated with \(L_3(2^m)\),” Algebra Logic 46 (5), 330–340 (2007).
D. V. Lytkina and V. D. Mazurov, “Groups containing a strongly embedded subgroup,” Algebra Logic 48 (2), 108–114 (2009).
D. V. Lytkina, “Groups saturated by finite simple groups,” Algebra Logic 48 (5), 357–370 (2009).
B. Li and D. V. Lytkina, “Sylow 2-subgroups of the periodic groups saturated with finite simple groups,” Siberian Math. J. 57 (6), 1029–1033 (2016).
D. V. Lytkina and A. A. Shlepkin, “Periodic groups saturated with finite simple groups of types \(U_3\) and \(L_3\),” Algebra Logic 55 (4), 289–294 (2016).
D. V. Lytkina and A. A. Shlepkin, “Periodic groups saturated with the linear groups of degree 2 and the unitary groups of degree 3 over finite fields of odd characteristic,” Siberian Adv. Math. 28 (3), 175–186 (2018).
D. V. Lytkina, A. I. Sozutov and A. A. Shlepkin, “2-rank two periodic groups saturated with finite simple groups,” Sib. Élektron. Mat. Izv. 15, 786–796 (2018).
Kourovka Notebook (Unsolved Problems in Group Theory) (Institute of Mathematics SO Russian Academy of Sciences, Novosibirsk, 2018) [in Russian].
A. A. Cherep, “The set of elements of finite order in a biprimitively finite group,” Algebra Logic 26 (4), 311–313 (1987).
B. Amberg and L. Kazarin, “Periodic groups saturated with dihedral subgroups,” International Algebraic Conference Dedicated to 70-th Birthday of Anatoly Yakovlev, Collection of Abstracts (Saint-Petersburg, 2010), pp. 79–80.
B. Amberg, A. Fransman, and L. Kazarin, “Products of locally dihedral subgroups,” J. Algebra 350, 308–317 (2013).
A. I. Starostin, “On Frobenius groups,” Ukrainian Math. J. 23 (5), 518–526 (1971 (1972)).
A. M. Popov, A. I. Sozutov, and V. P. Shunkov, Groups with Systems of Frobenius Subgroups (KGTU Press, Krasnoyarsk, 2004) [in Russian].
V. V. Bludov, “On Frobenius groups,” Siberian Math. J. 38 (6), 1054–1056 (1997).
A. I. Sozutov, “On the Shunkov groups acting freely on Abelian groups,” Siberian Math. J. 54 (1), 144–151 (2013).
A. Kh. Zhurtov, D. V. Lytkina, V. D. Mazurov, and A. I. Sozutov, “On periodic groups acting freely on Abelian groups,” Proc. Steklov Inst. Math. (Suppl. 1) 285, S209–S215 (2014).
S. I. Adyan, “Periodic products of groups,” Proc. Steklov Inst. Math 142, 1–19 (1979).
A. Yu. Ol’shanskii, Geometry of defining relations in groups (Kluwer Academic Publishers Group, Dordrecht, 1991).
A. I. Sozutov, “Groups with Frobenius pairs of conjugate elements,” Algebra Logic 16 (2), 136–143 (1978).
A. I. Sozutov, “Groups with finite Engel element,” Algebra Logic 58 (3), 254–267 (2019).
A. I. Sozutov, N. M. Suchkov, and N. G. Suchkova, Infinite Groups with Involutions (Siberian Federal Univ., Krasnoyarsk, 2011) [in Russian].
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This work was supported by the Russian Science Foundation under grant 19-71-10017.
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Sozutov, A.I. Groups Saturated with Finite Frobenius Groups. Math Notes 109, 270–279 (2021). https://doi.org/10.1134/S0001434621010314
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DOI: https://doi.org/10.1134/S0001434621010314