Cyclic Group
A cyclic group 
of Order 
is a Group defined by the element 
(theGenerator) and its Powers up to
where
is the Identity Element. Cyclic groups are both Abelian and Simple. There exists a unique cyclic group of every order 
, so cyclic groups of the same order are always isomorphic (Shanks 1993, p. 74), and all Groups of Prime Order are cyclic. Examples of cyclic groups include 
, 
, 
, and the Modulo Multiplication Groups 
such that 
, 4, 
, or 
, for 
an Odd Prime and 
(Shanks 1993, p. 92). Bycomputing the Characteristic Factors, any Abelian Group can be expressed as aDirect Product of cyclic Subgroups, for example, Finite Group Z2Z4 or Finite Group Z2Z2Z2.
References
Lomont, J. S. ``Cyclic Groups.'' §3.10.A in Applications of Finite Groups. New York: Dover, p. 78, 1987. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.
- Extremal Graph
- Extremals
- Extreme and Mean Ratio
- Extreme Value Distribution
- Extreme Value Theorem
- Extremum
- Extremum Test
- Extrinsic Curvature
- Fabry Imbedding
- Face
- Facet
- Faceting
- Factor
- Factor Base
- Factor (Graph)
- Factor Group
- Factorial
- Factorial Moment
- Factorial Number
- Factorial Prime
- Factorial Sum
- Factoring
- Factorion
- Factorization
- Factor Level