^{}Dipartimento di Matematica Universit`a di Salerno Via Ponte don Melillo 84084 - Fisciano (SA), Italy

Abstract

Many results were proved on the structure of finite groups with some restrictions on their real elements and on their conjugacy classes. We generalize a few of these to some classes of infinite groups. We study groups in which real elements are central, groups in which real elements are $2$-elements, groups in which all non-trivial classes have the same finite size and $FC$-groups with two non-trivial conjugacy class sizes.