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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.
Theorem 1 Every subgroup of an FC-group is an FC-group.
Theorem 3 Every restricted direct product of FC-groups is an FC-group.
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 quotient $G/\Tor(G)$ is a torsion-free abelian group.
Corollary 1 Every torsion-free FC-group is abelian.
Theorem 6 If $G$ is a finitely generated FC-group, then $G/Z(G)$ and $\Tor(G)$ 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.
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.
- 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.
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"FC-group" is owned by yark.
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| Also defines: |
FC, locally normal, locally normal group, locally finite and normal, locally finite and normal group, BFC-group, BFC group, BFC, finite-by-abelian, finite-by-abelian group, centre-by-finite group, center-by-finite group, central-by-finite group, centre-by-finite, center-by-finite, central-by-finite |
| Keywords: |
group, conjugacy |
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Cross-references: centre, contains, central quotient, normal subgroup, commutator subgroup, integer, finitely generated groups, stronger, normal closure, subset, subdirect product, finitely generated, abelian, torsion-free, order, periodic, homomorphic image, mnemonic, restricted direct products, abelian groups, finite groups, finite, centralizer, conjugates, element, group
This is version 19 of FC-group, born on 2004-12-10, modified 2009-08-12.
Object id is 6551, canonical name is FCGroup.
Accessed 13655 times total.
Classification:
| AMS MSC: | 20F24 (Group theory and generalizations :: Special aspects of infinite or finite groups :: FC-groups and their generalizations) |
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Pending Errata and Addenda
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