virtually cyclic group

A virtually cyclic group isa group that has a cyclic subgroup of finite index (http://planetmath.org/Coset).Every virtually cyclic group in facthas 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 ascyclic-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) groupsare 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.