About Periodic Shunkov Group Saturated with Finite Simple Groups of Lie Type Rank 1
The property of group G to be saturated with given set of groups X is a natural generalization of locally-cover definition (in class of locally finite groups) on periodic groups. Locally-finite group, witch has a locally-cover contains from finite simple Lie type groups of finite rank, is a Lie type group on some locally finite field. We call group ”Shunkov group” if every pair of conjugate elements generate finite subgroup, and this property saves after crossing on factor groups by finite subgroups. Group G saturated with groups from the set X, if every finite subgroup K from G contains in some subgroup G isomorphic to some group from X. In our work we solved the problem of building periodic Shunkov groups saturated with finite simple Lie groups of rank 1. Let M — is a set contains from finite simple groups Suzuki, Re, Unitary, Linear of Lie type rank 1. We proved that periodic Shunkov group saturated with groups from set M is isomorphic to simple group of Lie type rank 1 for some locally finite field Q. Also we got a description of Sylow 2-subgroup of periodic group saturated with groups from M, what is a necessary step in establishing of structure arbitrary periodic group with given saturation set.
1. Belyaev V.V. Locally finite Shevalle groups Explorations in group theory. Sverdlovsk, Ural science center AS USSR, 1984, pp. 39-50.
2. Kourovka notebook, Unsolved questions of group theory. 16 ed. Novosibirsk, Institute of mathematics SB RAS, 2006.
3. Kuznetcov A.A., Filippov K.A. Groups saturated with given set of groups. Siberian electronic mathematical reports, 2011. vol. 8, pp. 230-246.
4. Lytkina D.V. About groups saturated with finite simple groups. Algebra and logic, 2009, vol. 48 no 2, pp. 523-628.
5. Mazurov V.D. Periodic groups, saturated with L3(2m),. Algebra and logic, 2005, vol. 46, no 5, pp. 606–626.
6. Rubashkin A.G., Filippov K.A., About periodic groups saturated with groups L2(pn). Siberian mathematical journal, 2005, vol. 46 no 6, pp. 1388-1392.
7. Filippov K.A. About periodic groups saturated with finite simple groups. Siberian mathematical journal, 2012, vol. 52, no 2, pp. 430-438.
8. Shlepkin A.A. Periodic groups, saturated with wreath groups. Siberian electronic mathematical reports, 2013, vol. 10, pp. 56-64.
9. Shlepkin A.A. About periodic and Shunkov groups saturated with unitary groups of degree three.Proceedings of Institute of mathematics and mechanics Ural science center RAS, in print.
10. Amberg B., Kazarin L. Periodic groups saturated by dihedral subgroups. Book of abstracts of the international algebraic conference dedicated to 70-th birthday of Anatoly Yakovlev. Saint-Petersburg, 2010, pp. 79-80.
11. John N. Bray, Derek F. Holt, Colva M. Ronty – Dougal. The Maximal Subgroups of the Low – Dimensional Finite Classical groups. Cambridge university press, 2013, pp. 319–325.