A cyclic group of Order is a Group defined by the element (the
Generator) 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). By computing the Characteristic Factors, any Abelian Group can be expressed as a Direct 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.

© 1996-9

1999-05-25