Centralizers of finite subgroups in Hall’s universal group

Abstract

The structure of the centralizers of elements and finite abelian subgroups in Hall's universal group is studied by B. Hartley by using the property of existential closed structure of Hall’s universal group in the class of locally finite groups. The structure of the centralizers of arbitrary finite subgroups were an open question for a long time. Here by using basic group theory and the construction of P. Hall we give a complete description of the structure of centralizers of arbitrary finite subgroups in Hall's universal group. Namely we prove the following. Let $U$ be the Hall's universal group and $F$ be a finite subgroup of $U$.Then the centralizer $C_U(F)$ is isomorphic to an extention of $Z(F)$ by $U$.