On the Shunkov groups acting freely on Abelian groups.(English. Russian original)Zbl 1295.20039

Sib. Math. J. 54, No. 1, 144-151 (2013); translation from Sib. Mat. Zh. 54, No. 1, 188-198 (2013).
A group $$G$$ is called a Shunkov group if, for each finite subgroup $$F$$ of $$G$$, the subgroup generated by any two conjugate elements of prime order in the group $$N_G(F)/F$$ is finite. With Theorem 1 the author proves that the set of elements of finite order in a Shunkov group of rank $$1$$ (i.e. $$C_p\times C_p$$-free for all primes $$p$$) is a locally finite group. In Theorem 2 it is proved that the same conclusion holds for Shunkov groups acting regularly on Abelian groups (Theorem 2). These theorems generalize similar results by T. Grundhöfer and E. Jabara [Arch. Math. 97, No. 3, 219-223 (2011; Zbl 1241.20036)].
An interesting application of the above mentioned results is Corollary 1: If the group $$T_2(D)$$ of affine transformations of a neardomain $$D$$ is a Shunkov group and $$\mathrm{char}(D)\neq 2$$, then $$T_2(D)$$ has a locally finite periodic part and a regular elementary Abelian normal subgroup; furthermore $$D$$ is a nearfield of finite characteristic.

MSC:

 20F28 Automorphism groups of groups 20F50 Periodic groups; locally finite groups 20B22 Multiply transitive infinite groups 20E25 Local properties of groups 12K05 Near-fields

Zbl 1241.20036
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References:

 [1] Shunkov V. P., ”On a class of p-groups,” Algebra and Logic, 9, No. 4, 291–297 (1970). · Zbl 0237.20032 [2] Mazurov V., ”A new proof of Zassenhaus theorem on finite groups of fixed-point-free automorphisms,” J. Algebra, 263, No. 1, 1–7 (2003). · Zbl 1021.20013 [3] Zhurtov A. Kh., ”Frobenius groups containing the element of order 3,” Vladikavkaz. Mat. Zh., 2, No. 2, 19–25 (2000). · Zbl 1035.20033 [4] Zhurtov A. Kh., ”On regular automorphisms of order 3 and Frobenius pairs,” Siberian Math. J., 41, No. 2, 268–275 (2000). · Zbl 0956.20036 [5] Zhurtov A. Kh., ”On quadratic automorphisms of abelian groups,” Algebra and Logic, 39, No. 3, 184–188 (2000). · Zbl 0980.20049 [6] Mazurov V. D. and Churkin V. A., ”About a group that acts freely on an abelian group,” Siberian Math. J., 42, No. 4, 748–750 (2001). · Zbl 1016.20022 [7] Mazurov V. D. and Churkin V. A., ”On a free action of a group on an abelian group,” Siberian Math. J., 43, No. 3, 480–486 (2002). · Zbl 1009.20039 [8] Zhurtov A. Kh., ”Frobenius groups generated by two elements of order 3,” Siberian Math. J., 42, No. 3, 450–454 (2001). · Zbl 1003.20029 [9] Zhurtov A. Kh., ”On a group acting locally freely on an abelian group,” Siberian Math. J., 44, No. 2, 275–277 (2003). · Zbl 1035.20032 [10] Zhurtov A. Kh. and Mazurov V. D., ”Frobenius groups generated by quadratic elements,” Algebra and Logic, 42, No. 3, 153–164 (2003). · Zbl 1035.20034 [11] Mazurov V. D., ”A generalization of the Zassenhaus theorem,” Vladikavkaz. Mat. Zh., 10, No. 1, 40–52 (2008). · Zbl 1324.20007 [12] Lytkina D. V. and Mazurov V. D., ”Periodic groups generated by a pair of virtually quadratic automorphisms of an abelian group,” Siberian Math. J., 51, No. 3, 475–478 (2010). · Zbl 1207.20033 [13] Lytkina D. V., ”Periodic groups acting freely on Abelian groups,” Algebra and Logic, 49, No. 3, 256–261 (2010). · Zbl 1216.20026 [14] Grundhöfer T. and Jabara E., ”Fixed-point-free 2-finite automorphism groups,” Arch. Math., 97, 219–223 (2011). · Zbl 1241.20036 [15] Sozutov A. I., ”On the structure of the noninvariant factor in some Frobenius groups,” Siberian Math. J., 35, No. 4, 795–801 (1994). · Zbl 0851.20039 [16] Wähling H., Theorie der Fastkörper, Thalen Verlag, Essen (1987). · Zbl 0669.12014 [17] Mazurov V. D., ”Doubly-transitive permutation groups,” Siberian Math. J., 31, No. 4, 615–617 (1990). · Zbl 0742.20003 [18] Mazurov V. D., ”Sharply 2-transitive permutation groups,” in: Problems in Algebra and Logic [in Russian], Izdat. Inst. Mat., Novosibirsk, 1996, pp. 233–236. [19] Mazurov V. D. and Khukhro E. I. (Eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 17th ed., Sobolev Inst. Math., Novosibirsk (2010). · Zbl 1211.20001 [20] Ostylovskiĭ A. N. and Shunkov V. P., ”On the local finiteness of a certain class of groups with the minimality condition for subgroups,” in: Studies in Group Theory [in Russian], Inst. Fiz. Sibirsk. Otdel. Akad. Nauk SSSR, Krasnoyarsk, 1975, pp. 32–48. [21] Hall M., Jr., The Theory of Groups [Russian translation], Chelsea Publishing, Providence, RI (1998). [22] Cherep A. A., ”The set of elements of finite order in a biprimitively finite group,” Algebra and Logic, 26, No. 4, 311–313 (1987). · Zbl 0663.20029 [23] Shunkov V. P., ”A test for the nonsimplicity of groups,” Algebra i Logika, 14, No. 5, 491–522 (1975). · Zbl 0382.20018 [24] Busarkin V. M. and Gorchakov Yu. M., Decomposable Finite Groups [in Russian], Nauka, Moscow (1968). [25] Starostin A. I., ”On Frobenius groups,” Ukrain. Mat. Zh., 23, No. 5, 629–639 (1971). · Zbl 0224.20014 [26] Sozutov A. I. and Shunkov V. P., ”On a generalization of Frobenius’ theorem to infinite groups,” Math. USSR-Sb., 29, No. 4, 441–451 (1976). · Zbl 0376.20022 [27] Popov A. M. and Sozutov A. I., ”A group with H-Frobenius element of even order,” Algebra and Logic, 44, No. 1, 40–45 (2005). · Zbl 1079.20056
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