A virtually cyclic group is a group that has a cyclic subgroup of finite index. 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-$\Z$ , in which case the extension splits.

Finite-by-dihedral groups are also virtually cyclic. In fact, we have the following classification theorem:[1][2]

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

Bibliography

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).