Groups with Restrictions on Infinite Conjugacy Classes

A group G is said to be an AFC-group if for each element x of G at least one of the indices |CG(x):⟨x⟩|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|C_G(x):\langle x\rangle |$$\end{document} and |G:CG(x)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|G:C_G(x)|$$\end{document} is finite. Groups with this property appear as a natural generalization of those with finite conjugacy classes, and the aim of this paper is to investigate the structure of AFC-groups, and in particular the behaviour of their FC-centre.