Virtually Cyclic Groups
A group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of cosets that the subgroup has). In other words, any element in a virtually cyclic group can be arrived at by applying a member of the cyclic subgroup to a member in a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. It is known that a finitely generated discrete group with exactly two ends is virtually cyclic (for instance the product of Z/n and Z). Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.
Read more about this topic: Cyclic Group
Famous quotes containing the words virtually and/or groups:
“If twins are believed to be less intelligent as a class than single-born children, it is not surprising that many times they are also seen as ripe for social and academic problems in school. No one knows the extent to which these kind of attitudes affect the behavior of multiples in school, and virtually nothing is known from a research point of view about social behavior of twins over the age of six or seven, because this hasn’t been studied either.”
—Pamela Patrick Novotny (20th century)
“Only the groups which exclude us have magic.”
—Mason Cooley (b. 1927)