# 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^{}.

## 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-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 |