Definition
Cyclic Group
A cyclic group is a group
that can be generated by a single element. There exists an element , called a generator, such that every element can be written as a power (or multiple) of : where
denotes the -fold application of the group operation.
Finite and Infinite Cyclic Groups
Finite Cyclic Groups
If
has finite order , then is isomorphic to the additive group of integers modulo : The generator
satisfies , and the elements are .
Infinite Cyclic Groups
If
has infinite order, then is isomorphic to the additive group of integers:
Properties
Abelian
Subgroups
Every subgroup of a cyclic group is cyclic.
If
has finite order , then for each divisor of there exists exactly one subgroup of order , namely .
Generators
Let
be a finite cyclic group of order . An element is a generator of iff and are coprime. Consequently, the number of distinct generators of
is , where denotes Euler’s totient function.
Examples
Integers Modulo n
is cyclic for every . The element is always a generator.
Multiplicative Group of a Finite Field
For a prime
, the multiplicative group is cyclic of order . Its generators are exactly the primitive roots modulo .
Trivial Group
The group
consisting of only the identity element is cyclic, generated by .
Relation to Discrete Logarithm
Discrete Logarithm
In a finite cyclic group
of order , every element can be written uniquely as for some . The exponent is the discrete logarithm of with respect to . This underlies the security of Diffie-Hellman and ElGamal.