FC-group

An FC-group is a group in which every element has only finitely many conjugates. Equivalently, a group $G$ is an FC-group if and only if the centralizer $C_G(x)$ is of finite index in $G$ for each $x\in G$ .

All finite groups and all abelian groups are obviously FC-groups. Further examples of FC-groups can be obtained by taking restricted direct products of such groups.

The term FC-group was introduced by Baer[1]; the FC is simply a mnemonic for the definition involving finite conjugacy classes.

Some theorems

From Theorem 4 above it follows that a group $G$ is a periodic FC-group if and only if every finite subset of $G$ has a finite normal closure. For this reason, periodic FC-groups are sometimes called locally normal (or locally finite and normal) groups.

Stronger properties

The following two properties are sometimes encountered, both of which are somewhat stronger than being an FC-group. For finitely generated groups they are in fact equivalent to being an FC-group, by Theorem 6 above.

A BFC-group is a group $G$ such that every conjugacy class of elements of $G$ has at most $n$ elements, for some fixed integer $n$ . B. H. Neumann showed[2] that $G$ is a BFC-group if and only if its commutator subgroup $[G,G]$ is finite (which in turn is easily shown to be equivalent to $G$ being finite-by-abelian, that is, having a finite normal subgroup $N$ such that $G/N$ is abelian).

A centre-by-finite (or central-by-finite) group is a group $G$ such that the central quotient $G/Z(G)$ is finite. A centre-by-finite group is necessarily a BFC-group, because the centralizer of any element contains the centre.

References

1

R. Baer, Finiteness properties of groups, Duke Math. J. 15 (1948), 1021-1032.

2

B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236-248.