Editing Cyclic group (section)

===Virtually cyclic groups===
A group is called '''virtually cyclic''' if it contains a cyclic subgroup of finite [[index (group theory)|index]] (the number of [[coset]]s that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by multiplying a member of the cyclic subgroup and a member of a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is [[finitely generated group|finitely generated]] and has exactly two [[End (graph theory)|ends]];{{refn|group=note|If ''G'' has two ends, the explicit structure of ''G'' is well known: ''G'' is an extension of a finite group by either the infinite cyclic group or the infinite dihedral group.<ref>{{Harv|Stallings|1970|pp=124–128}}. See in particular {{Google books|3Lyvsc694T4C|Groups of cohomological dimension one|page=126}}.</ref>}} an example of such a group is the [[direct product of groups|direct product]] of '''Z'''/''n'''''Z''' and '''Z''', in which the factor '''Z''' has finite index&nbsp;''n''. Every abelian subgroup of a [[hyperbolic group|Gromov hyperbolic group]] is virtually cyclic.<ref>{{Harv|Alonso|1991|loc = Corollary&nbsp;3.6}}.</ref>