Cyclic Groups


Cyclic groups are one of the key building blocks for groups - especially abelian groups. The finite cyclic groups take several common forms including the integers mod n for positive integers n. This is denoted by Z/nZ, where Z is the ring of integers.


A finite group G is called a cyclic group if it is generated by a single element. The way to write this symbolically is G = <g>, where the notation <g> stands for all powers of g. (Don't forget, even though we'll sometimes use words like 'powers' which sounds like we're talking about multiplication, we are talking about a generic group.) Another way to think about this is that if G is cyclic, and g is a generator of order n, then all the elements in the group are {e, g, g2, g3, ..., gn-1}. The word cyclic is used because the sequence g, g^2, g^3, ... eventually cycles around and begins to repeat.

There are two types of cyclic groups: the integers Z under addition, and the integers mod n under addition.


If you look at the solutions to the equation xn-1=0, you get what's called the "nth roots of unity." If you multiply one nth root of unity by another, you get a third nth root of unity. So the nth roots of unity form a group under multiplication. It is a cyclic group because the complex number e2πi/n generates the group.