FC-group

locally finite

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.

1 Some theorems

Theorem 1.

Every subgroup (http://planetmath.org/Subgroup) of an FC-group is an FC-group.

Theorem 2.

Every homomorphic image of an FC-group is an FC-group.

Theorem 3.

Every restricted direct product of FC-groups is an FC-group.

Theorem 4.

Every periodic FC-group is locally finite (http://planetmath.org/LocallyFiniteGroup).

Theorem 5.

Let $G$ be an FC-group.The elements of finite order in $G$ form a subgroup,which will be denoted by $\mathrm{Tor}\mathit{}\mathrm{(}G\mathrm{)}$.The subgroup $\mathrm{Tor}\mathit{}\mathrm{(}G\mathrm{)}$ is a periodic FC-group,and the quotient (http://planetmath.org/QuotientGroup) $G\mathrm{/}\mathrm{Tor}\mathit{}\mathrm{(}G\mathrm{)}$ is a torsion-free abelian group.

Corollary 1.

Every torsion-free FC-group is abelian.

Theorem 6.

If $G$ is a finitely generated FC-group,then $G\mathrm{/}Z\mathit{}\mathrm{(}G\mathrm{)}$ and $\mathrm{Tor}\mathit{}\mathrm{(}G\mathrm{)}$ are both finite.

Theorem 7.

Every FC-group is a subdirect product of a periodic FC-groupand a torsion-free abelian group.

From Theorem 4 above it follows that a group $G$ is a periodic FC-groupif 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) groupis 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.