# Cyclic groups

## Defining Cyclic Groups

• A cyclic group is a special type of group that can be generated by a single element.
• The element that generates a cyclic group is called a generator of the group.
• These groups are called ‘cyclic’ because their elements can be listed in a cycle.
• Every cyclic group is abelian, which means its operation is commutative - the order in which elements are combined does not matter.

## Properties of Cyclic Groups

• For a group to be cyclic, there must exist an element ‘g’ in the group such that every other element of the group can be written as a power of ‘g’.
• The order of an element is defined as the least positive integer ‘n’ such that g^n equals the identity element. It is also the size (number of elements) of the cyclic subgroup generated by that element.
• The order of a group is the number of elements in the group. If the group is finite and has ‘n’ elements, the group is said to have order ‘n’.

## Infinite and Finite Cyclic Groups

• Cyclic groups can be finite or infinite. If the powers of ‘g’ eventually repeat, the group is finite, otherwise, it is infinite.
• In a finite cyclic group of order ‘n’, there is a unique subgroup of order ‘d’ for every divisor ‘d’ of ‘n’, no two subgroups of the same order are the same.
• For infinite cyclic groups, each non-identity element generates a copy of the group of integers, under addition, and so has infinite order.
• An infinite cyclic group has exactly two generators, one for the positive integers and one for the negatives.

## Examples of Cyclic Groups

• The integers Z under addition, denoted (Z, +), form an infinite cyclic group. The generator is either 1 or -1.
• A finite cyclic group example is the clock arithmetic, also known as the integers mod n, denoted Z_n. It has n elements {0, 1, …, n-1} and the operation is addition modulo n. This group is generated by 1.
• Another example is the complex nth roots of unity under multiplication. It’s a finite group, and its elements are e^(2πik/n) for k=0,…,n-1.