# virtually cyclic group

A virtually cyclic group is a group that has a cyclic subgroup of finite index (http://planetmath.org/Coset). Every virtually cyclic group in fact has a normal cyclic subgroup of finite index (namely, the core of any cyclic subgroup of finite index), and virtually cyclic groups are therefore also known as cyclic-by-finite groups.

A finite-by-cyclic group (that is, a group $G$ with a finite normal subgroup $N$ such that $G/N$ is cyclic) is always virtually cyclic. To see this, note that a finite-by-cyclic group is either finite, in which case it is certainly virtually cyclic, or it is finite-by-$\mathbb{Z}$, in which case the extension (http://planetmath.org/GroupExtension) splits (http://planetmath.org/SemidirectProductOfGroups).

Finite-by-dihedral (http://planetmath.org/DihedralGroup) groups are also virtually cyclic. In fact, we have the following classification theorem:[1][2]

###### Theorem.

Groups of the following three types are all virtually cyclic. Moreover, every virtually cyclic group is of exactly one of these three types.

As an immediate corollary we have the following result:[3]

###### Corollary.

Every torsion-free virtually cyclic group is either trivial or infinite cyclic.

## References

• 1 Lemma 11.4 (pages 102–103) in: John Hempel, 3-Manifolds, American Mathematical Society, 2004, ISBN 0821836951.
• 2 Page 137 of: Alejandro Adem, Jesus Gonzalez, Guillermo Pastor (eds.), Recent developments in algebraic topology — A conference to celebrate Sam Gitler’s 70th birthday, San Miguel de Allende, Mexico, December 3–6, 2003.
• 3 Lemma 3.2 (pages 225–226) of: Dugald Macpherson, Permutation Groups Whose Subgroups Have Just Finitely Many Orbits (pages 221–230 in: W. Charles Holland (ed.) Ordered Groups and Infinite Permutation Groups, Kluwer Academic Publishers, 1996).
 Title virtually cyclic group Canonical name VirtuallyCyclicGroup Date of creation 2013-03-22 15:47:15 Last modified on 2013-03-22 15:47:15 Owner yark (2760) Last modified by yark (2760) Numerical id 11 Author yark (2760) Entry type Definition Classification msc 20F19 Classification msc 20E22 Synonym cyclic-by-finite group Related topic VirtuallyAbelian Defines virtually cyclic Defines cyclic-by-finite Defines finite-by-cyclic group Defines finite-by-cyclic