FC-group

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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 $\operatorname{Tor}(G)$ . The subgroup $\operatorname{Tor}(G)$ is a periodic FC-group, and the quotient (http://planetmath.org/QuotientGroup) $G/\operatorname{Tor}(G)$ 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 $G/Z(G)$ and $\operatorname{Tor}(G)$ are both finite.

Theorem 7.

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

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.

Title	FC-group
Canonical name	FCgroup
Date of creation	2013-03-22 14:52:28
Last modified on	2013-03-22 14:52:28
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	22
Author	yark (2760)
Entry type	Definition
Classification	msc 20F24
Synonym	FC group
Defines	FC
Defines	locally normal
Defines	locally normal group
Defines	locally finite and normal
Defines	locally finite and normal group
Defines	BFC-group
Defines	BFC group
Defines	BFC
Defines	finite-by-abelian
Defines	finite-by-abelian group
Defines	centre-by-finite group
Defines	center-by-finite group
Defines	central-by-finite group
Defines	centre-by