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Difference between revisions of "FC-group"

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''finite conjugate group''
 
''finite conjugate group''
  
A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300301.png" /> such that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300302.png" /> has only finitely many conjugates. This is one of several important possible finiteness conditions on an (infinite) group (cf. also [[Group with a finiteness condition|Group with a finiteness condition]]). FC-groups are similar to finite groups in several respects.
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A [[Group|group]] $G$ such that each $x\in G$ has only finitely many conjugates. This is one of several important possible finiteness conditions on an (infinite) group (cf. also [[Group with a finiteness condition|Group with a finiteness condition]]). FC-groups are similar to finite groups in several respects.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300303.png" /> be an arbitrary group. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300304.png" /> is an FC-element if it has only finitely many conjugates. The FC-elements form a characteristic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300305.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300306.png" /> is residually finite (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300307.png" /> is the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300308.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f1300309.png" />).
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Let $G$ be an arbitrary group. An element $x\in G$ is an FC-element if it has only finitely many conjugates. The FC-elements form a characteristic subgroup $F$, and $G/C_G(F)$ is residually finite (here, $C_G(F)$ is the centralizer of $F$ in $G$).
  
 
An FC-group is thus a group in which all elements are FC-elements.
 
An FC-group is thus a group in which all elements are FC-elements.
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The commutator subgroup of an FC-group is periodic (torsion).
 
The commutator subgroup of an FC-group is periodic (torsion).
  
A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f13003010.png" /> is a finitely-generated FC-group if and only if it has a free Abelian subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f13003011.png" /> of finite rank in its centre such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f13003012.png" /> is of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130030/f13003013.png" />.
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A group $G$ is a finitely-generated FC-group if and only if it has a free Abelian subgroup $A$ of finite rank in its centre such that $A$ is of finite index in $G$.
  
 
For further results, see [[#References|[a1]]], Part 1, Sect. 4.3; Part 2, pp. 102–104, and [[#References|[a2]]], Sect. 15.1. See also [[CC-group|CC-group]].
 
For further results, see [[#References|[a1]]], Part 1, Sect. 4.3; Part 2, pp. 102–104, and [[#References|[a2]]], Sect. 15.1. See also [[CC-group|CC-group]].

Latest revision as of 21:41, 30 April 2014

finite conjugate group

A group $G$ such that each $x\in G$ has only finitely many conjugates. This is one of several important possible finiteness conditions on an (infinite) group (cf. also Group with a finiteness condition). FC-groups are similar to finite groups in several respects.

Let $G$ be an arbitrary group. An element $x\in G$ is an FC-element if it has only finitely many conjugates. The FC-elements form a characteristic subgroup $F$, and $G/C_G(F)$ is residually finite (here, $C_G(F)$ is the centralizer of $F$ in $G$).

An FC-group is thus a group in which all elements are FC-elements.

The commutator subgroup of an FC-group is periodic (torsion).

A group $G$ is a finitely-generated FC-group if and only if it has a free Abelian subgroup $A$ of finite rank in its centre such that $A$ is of finite index in $G$.

For further results, see [a1], Part 1, Sect. 4.3; Part 2, pp. 102–104, and [a2], Sect. 15.1. See also CC-group.

References

[a1] D.J.S. Robinson, "Finiteness conditions and generalized soluble groups, Parts 1–2" , Springer (1972)
[a2] W.R. Scott, "Group theory" , Dover, reprint (1987)
How to Cite This Entry:
FC-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=FC-group&oldid=32010
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article