A group $G$ is saturated with groups from a set of groups $\mathfrak{X}$ if any finite subgroup $K$ of $G$ is contained in a subgroup of $G$ isomorphic to some group from~$\mathfrak{X}$. A group $G$ is called a Shunkov group (a conjugately biprimitively finite group) if, for any finite subgroup $H$ of $G$, any two conjugate elements of prime order in the quotient group $N_G(H)/h$ generate a finite group. Let $G$ be a group. If all elements of finite orders from $G$ are contained in a periodic subgroup of $G$, then it is called a periodic part of $G$ and is denoted by $t(G)$. It is known that a Shunkov group may have no periodic part. The existence of a periodic part of a Shunkov group saturated with finite wreathed groups is proved and the structure of the periodic part is established.
Keywords: group saturated with a set of groups, Shunkov group
The paper was received by the Editorial Office on Juny 5, 2018.
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-31-00257).
Aleksei Anatolievich Shlepkin, Cand. Sci (Phys.-Math.), Institute of Space and Information Technologies of Siberian Federal University, Krasnoyarsk, 660074 Russia, e-mail: shlyopkin@gmail.com
REFERENCES
1. Shlepkin A.K. Conjugately biprimitive finite groups containing finite unsolvable subgroups. In: Abstracts. III International conf. on algebra (Krasnoyarsk, 1993), p. 369 (in Russian).
2. Senashov V.I., Shunkov V.P. Gruppy s usloviyami konechnosti. [Groups with finiteness conditions]. Novosibirsk: Izdatel’stvo Rossiiskoi Akademii Nauk, Sibirskoe Otdelenie, 2001, 336 p.
ISBN: 5-7692-0439-7 .
3. Cherep A.A. Set of elements of finite order in a biprimitively finite group. Algebra and Logic, 1987, vol. 26, no. 4, pp. 311–313. doi: 10.1007/BF01980245 .
4. Kargapolov M.I., Merzljakov Ju.I. Fundamentals of the theory of groups. N Y; Heidelberg; Berlin: Springer-Verlag, 1979, Ser. Graduate Texts in Mathematics, vol. 62, 203 p. ISBN: 978-1-4612-9966-0 . Original Russian text (3rd ed.) published in Kargapolov M.I., Merzlyakov Yu.I. Osnovy teorii grupp, Moscow, Nauka Publ., 1982, 288 p.
5. Shlepkin A.A. Periodic groups, saturated by wreathed groups. Sib. Elektron. Mat. Izv., 2013, vol. 10, pp. 56–64 (in Russian) .
6. Ditsman A.P. On the center of p-groups. Proceedings of the seminar on group theory. Moscow, 1938, pp. 30–34.
7. Shlepkin A.A. On Shunkov groups, saturated with linear and unitary groups of dimension 3 over fields of odd orders. Sib. Elektron. Mat. Izv., 2016, vol. 13, pp. 341–351 (in Russian) . doi: 10.17377/semi.2016.13.029 .
8. Shlepkin A.K. Gruppy Shunkova s dopolnitel’nymi ogranicheniyami. doktorskaya dissertatsiya. [Shunkov groups with additional restrictions. Doctoral dissertation]. Krasnoyarsk, 1999, 187 p.
9. Shlepkin A.K. Conjugately biprimitively finite groups with the primary minimal condition. Algebra Logic, 1983, vol. 22, no. 2, pp. 165–169. doi: 10.1007/BF01978669 .
10. Lytkina D.V., Tukhvatullina L.R., Filippov K.A. Sib. Math. J., 2008, vol. 49, no. 2, pp. 317–321. doi: 10.1007/s11202-008-0031-y .
