On Periodic Groups and Shunkov Groups that are Saturated by Dihedral Groups and A5
A group is said to be periodic, if any of its elements is of finite order. A Shunkov group is a group in which any pair of conjugate elements generates Finite subgroup with preservation of this property when passing to factor groups by finite Subgroups. The group G is saturated with groups from the set of groups X if any A finite subgroup K of G is contained in the subgroup of G, Isomorphic to some group in X. The paper establishes the structure of periodic groups And Shunkov groups saturated by the set of groups M consisting of one finite simple non-Abelian group A5 and dihedral groups with Sylow 2-subgroup of order 2. It is proved that A periodic group saturated with groups from M, is either isomorphic to a prime Group A5, or is isomorphic to a locally dihedral group with Sylow 2 subgroup of order 2. Also, the existence of the periodic part of the Shunkov group saturated with groups from the set M is proved, and the structure of this periodic part is established.
1. Ditsman A.P. On the center of p-groups. In Sat. Proceedings of the seminar on group theory. Moscow, 1938, pp. 30-34.
2. Kargapolov M.И., Merzlyakov Yu.I. Fundamentals of group theory. Moscow, Nauka Publ., 1982.
3. Kuznetsov A.A., Filippov К.А. Groups. Saturated with given Set of groups. Siberian electronic mathematical repots, 2011, vol. 8, pp. 230-246.
4. Mazurov V.D. On infinite groups with Abelian centralizers of involutions. Algebra and logic, 2009, vol. 39, no 1, pp. 74-86.
5. Rubashkin A.G. Groups saturated with given sets of finite groups. PhD thesis. 2005.
6. Filippov K.A. On the periodic part of the Shunkov group saturated with L2(pn). Bulletin of SibSAU, 2012, no 1, pp. 611-617.
7. Filippov K.A. On periodic groups saturated by finite simple groups. Sib. mat. Journal, 2012, vol. 53, no 2, pp. 430-438.
8. Ostylovsky A.N., Shunkov V.P. On the local finiteness of a class of groups with the minimality condition. Studies on the theory of groups. Krasnoyarsk, 1975, pp. 32-48.
9. Senashov V.I., Shunkov V.P. Groups with finiteness conditions. Novosibirsk, Izdatel’stvo SB RAN, 2001.
10. Shlepkin A.K. On conjugately biprimitively finite groups with a primary minimum condition, Algebra and logic, 1983, no 22, pp. 226-231.
11. Shlepkin A.K. Conjugately biprimitively finite groups containing finite unsolvable subgroups. Third Intern. Conf. In algebra. Krasnoyarsk, 1993.
12. Shlepkin A.K. Shunkov groups with additional restrictions. Thesis doc. Phys. Sciences. Krasnoyarsk State Univer, 1999.
13. Shlepkin A.K., Rubashkin A.G. On a class of periodic groups. Algebra and Logic, 2005, no 1, pp. 114-125.
14. Shunkov V.P. On periodic groups with almost regular involution. Algebra and logic, 1972, no 4, pp. 470-494.
15. A.A. Skull, On elements of finite order in biprimitively finite groups. Algebra and logic, 1987, no 26, pp. 518-521.
16. Amberg B., Kazarin L. Periodic groups saturated by dihedral subgroups. Book of abstracts of the international algebraic conference dedicated to the 70th birthday of Anatoly Yakovlev. Saint-Petersburg, 2010, pp. 79-80.