cyclic group
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 paragraphimply 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.)
- •
Every finite subgroup of the multiplicative group
of a field is cyclic.
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 |