Abstract
A group G is said to be an AFC-group if for each element x of G at least one of the indices \(|C_G(x):\langle x\rangle |\) and \(|G:C_G(x)|\) 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.
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