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 with a finite normal subgroup such that 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-, in which case the extension (http://planetmath.org/GroupExtension) splits (http://planetmath.org/SemidirectProductOfGroups).
Groups of the following three types are all virtually cyclic. Moreover, every virtually cyclic group is of exactly one of these three types.
Every torsion-free virtually cyclic group is either trivial or infinite cyclic.
- 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).
|Title||virtually cyclic group|
|Date of creation||2013-03-22 15:47:15|
|Last modified on||2013-03-22 15:47:15|
|Last modified by||yark (2760)|