On a Sufficient Condition for the Existence of a Periodic Part in the Shunkov Group

Abstract

The group $ G $ is saturated with groups from the set of groups if any a finite subgroup $ K $ of $ G $ is contained in a subgroup of $ G $, which is isomorphic to some group in $ \mathfrak{X} $. The set $ \mathfrak{X} $ from the above definition is called the saturating set for the group. By the Shunkov group $ G $ we mean a group in which for any of its finite subgroup $ H $ in the factor group $ N_G (H) / H $ any two conjugate elements of prime order generate a finite subgroup. The Shunkov group does not have to be periodic. Therefore, the problem of the location of elements of finite order in the Shunkov group with the saturation condition must be solved separately. If in a group $ G $ all elements of finite orders are contained in a periodic subgroup of the group $ G $, then it is called the periodic part of the group $ G $. It was proved that a periodic Shunkov group, saturated with finite simple non-abelian groups of Lie type of rank 1, is isomorphic to a group of Lie type of rank 1 over a suitable locally finite field. In this paper we consider arbitrary Shunkov groups (not necessarily periodic). It is proved that the Shunkov group $ G $, saturated with groups from the set of finite simple groups of Lie type of rank $ 1 $, has a periodic part that is isomorphic to a simple group of Lie type of rank $ 1 $ over a sutable locally finite field.

Keywords

• groups saturated with the set of groups
• Shunkov group