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 cyclicbyfinite groups.
A finitebycyclic 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 finitebycyclic group is either finite, in which case it is certainly virtually cyclic, or it is finiteby$\mathbb{Z}$, in which case the extension (http://planetmath.org/GroupExtension) splits (http://planetmath.org/SemidirectProductOfGroups).
Finitebydihedral (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.

•
finite

•
finiteby(infinite cyclic)

•
finiteby(infinite dihedral)
Corollary.
Every torsionfree virtually cyclic group is either trivial or infinite cyclic.
References
 1 Lemma 11.4 (pages 102–103) in: John Hempel, 3Manifolds, 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  20130322 15:47:15 
Last modified on  20130322 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  cyclicbyfinite group 
Related topic  VirtuallyAbelian 
Defines  virtually cyclic 
Defines  cyclicbyfinite 
Defines  finitebycyclic group 
Defines  finitebycyclic 