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.
- Pocklington's Criterion
- Pocklington's Theorem
- Poggendorff Illusion
- Pohlke's Theorem
- Poincaré-Birkhoff Fixed Point Theorem
- Poincaré Conjecture
- Poincaré Duality
- Poincaré Formula
- Poincaré-Fuchs-Klein Automorphic Function
- Poincaré Group
- Poincaré-Hopf Index Theorem
- Poincaré Hyperbolic Disk
- Poincaré Manifold
- Poincaré Metric
- Poincaré Separation Theorem
- Poincaré's Holomorphic Lemma
- Poincaré's Lemma
- Poinsot Solid
- Poinsot's Spirals
- Point
- Point at Infinity
- Point Estimator
- Point Groups
- Point-Line Distance--2-D
- Point-Line Distance--3-D