A group is said to be cyclic if it is generated by a single element.
Suppose is a cyclic group generated by . Then every element of is equal to for some . If is infinite, then these are all distinct, and is isomorphic to the group . If has finite order (http://planetmath.org/OrderGroup) , then every element of can be expressed as with , and is isomorphic to the quotient group .
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 , and the finite cyclic group of order is sometimes written . However, when the cyclic groups are written additively, they are commonly represented by and .
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 generator: an infinite cyclic group has generators, and a finite cyclic group of order has generators, where is the Euler totient function.
Some basic facts about cyclic groups:
Every cyclic group is abelian.
Every subgroup of a cyclic group is cyclic.
Every quotient of a cyclic group is cyclic.
Every group of prime order is cyclic. (This follows immediately from Lagrange’s Theorem.)
|Date of creation||2013-03-22 12:23:27|
|Last modified on||2013-03-22 12:23:27|
|Last modified by||yark (2760)|
|Defines||infinite cyclic group|
|Defines||infinite cyclic subgroup|