Centralizers in Hall's universal and direct limits of finitary symmetric groups

Abstract

We study the centralizers of finite subgroups in Hall's universal group. We describe the structure of the centralizers of arbitrary finite subgroups in the groups $S(\xi)$ and $FSym(\kappa)(\xi)$ where $S(\xi)$'s are obtained as direct limits of finite symmetric groups and $FSym(\kappa)(\xi)$'s are obtained as direct limits of finitary symmetric groups on the set of infinite cardinality $\kappa$.