locally finitePlanetmathPlanetmath

An FC-group is a group in which every element has only finitely many conjugatesPlanetmathPlanetmathPlanetmath. Equivalently, a group G is an FC-group if and only if the centralizerMathworldPlanetmath CG(x) is of finite index in G for each xG.

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

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

1 Some theorems

Theorem 1.

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

Theorem 2.

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

Theorem 3.

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

Theorem 4.

Every periodicPlanetmathPlanetmath 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 Tor(G). The subgroup Tor(G) is a periodic FC-group, and the quotientPlanetmathPlanetmath (http://planetmath.org/QuotientGroup) G/Tor(G) is a torsion-free abelian group.

Corollary 1.

Every torsion-free FC-group is abelian.

Theorem 6.

If G is a finitely generatedMathworldPlanetmathPlanetmath FC-group, then G/Z(G) and Tor(G) are both finite.

Theorem 7.

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

From TheoremMathworldPlanetmath 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 closurePlanetmathPlanetmath. 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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 subgroupMathworldPlanetmath [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 subgroupMathworldPlanetmath 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.


  • 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