A cyclic group is a group that can be generated by a single element 
(the group generator). Cyclic groups are Abelian.
A cyclic group of finite group order 
is denoted 
, 
,
, or 
; Shanks 1993, p. 75), and its generator 
satisfies
 |
(1)
|
where 
is the identity element.
The ring of integers 
form an infinite cyclic group under addition, and the integers
0, 1, 2, ..., 
(
) form a cyclic group of order 
under addition (mod 
). In both cases, 0 is the identity
element.
There exists a unique cyclic group of every order 
, so cyclic groups of the same order are always isomorphic
(Scott 1987, p. 34; Shanks 1993, p. 74). Furthermore, subgroups of cyclic
groups are cyclic, and all groups of prime group order are cyclic. In fact, the only simple Abelian groups are the cyclic groups of order 
or 
a prime (Scott 1987, p. 35).
The 
th cyclic group is represented in the
Wolfram Language as CyclicGroup[n].
Examples of cyclic groups include 
, 
,
, ..., and the modulo
multiplication groups 
such that 
,
4, 
, or 
, for 
an odd prime and 
(Shanks 1993, p. 92).
Cyclic groups all have the same multiplication table structure. The table for 
is illustrated above.
By computing the characteristic factors, any Abelian group can be expressed as a group
direct product of cyclic subgroups, for example,
finite group C2×C4 or finite
group C2×C2×C2. It is common to combine the indices for the highest
prime factors of the direct product representation of a group since this provides
a shorter notation and no ambiguity arises. For example 
is commonly written 
.
The cycle index of the cyclic group 
is given by
 |
(2)
|
where 
means 
divides 
and 
is the totient function
(Harary 1994, p. 184). The first few are given by
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
