FC-group

finite conjugate group

A group $G$ such that each $x\in G$ has only finitely many conjugates. This is one of several important possible finiteness conditions on an (infinite) group (cf. also Group with a finiteness condition). FC-groups are similar to finite groups in several respects.

Let $G$ be an arbitrary group. An element $x\in G$ is an FC-element if it has only finitely many conjugates. The FC-elements form a characteristic subgroup $F$, and $G/C_G(F)$ is residually finite (here, $C_G(F)$ is the centralizer of $F$ in $G$).

An FC-group is thus a group in which all elements are FC-elements.

The commutator subgroup of an FC-group is periodic (torsion).

A group $G$ is a finitely-generated FC-group if and only if it has a free Abelian subgroup $A$ of finite rank in its centre such that $A$ is of finite index in $G$.

For further results, see [a1], Part 1, Sect. 4.3; Part 2, pp. 102–104, and [a2], Sect. 15.1. See also CC-group.

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