Groups with reality and conjugacy conditions

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.