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.

• finite

• finite-by-(infinite cyclic)

• finite-by-(infinite dihedral)

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

Corollary.

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

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