cyclic group

A group is said to be cyclic if it is generated by a single element.

Suppose G is a cyclic groupMathworldPlanetmath generated by xG. Then every element of G is equal to xk for some k. If G is infinite, then these xk are all distinct, and G is isomorphicPlanetmathPlanetmathPlanetmath to the group . If G has finite order ( n, then every element of G can be expressed as xk with k{0,,n-1}, and G is isomorphic to the quotient groupMathworldPlanetmath /n.

Note that the isomorphisms mentioned in the previous paragraph imply that all cyclic groups of the same order are isomorphic to one another. The infinite cyclic group is sometimes written C, and the finite cyclic group of order n is sometimes written Cn. However, when the cyclic groups are written additively, they are commonly represented by and /n.

While a cyclic group can, by definition, be generated by a single element, there are often a number of different elements that can be used as the generatorPlanetmathPlanetmathPlanetmath: an infinite cyclic group has 2 generators, and a finite cyclic group of order n has ϕ(n) generators, where ϕ is the Euler totient function.

Some basic facts about cyclic groups:

Title cyclic group
Canonical name CyclicGroup
Date of creation 2013-03-22 12:23:27
Last modified on 2013-03-22 12:23:27
Owner yark (2760)
Last modified by yark (2760)
Numerical id 21
Author yark (2760)
Entry type Definition
Classification msc 20A05
Related topic GeneralizedCyclicGroup
Related topic PolycyclicGroup
Related topic VirtuallyCyclicGroup
Related topic CyclicRing3
Defines cyclic
Defines cyclic subgroup
Defines infinite cyclic
Defines infinite cyclic group
Defines infinite cyclic subgroup