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 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.)
-
•
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 |