Известия Иркутского государственного университета: Серия "Математика" (Jun 2016)
About Periodic Shunkov Group Saturated with Finite Simple Groups of Lie Type Rank 1
Abstract
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 $\mathfrak{M}$ --- 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 $\mathfrak{M}$ 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 $\mathfrak{M},$ what is a necessary step in establishing of structure arbitrary periodic group with given saturation set.
