Cyclic groups – A Level Further Mathematics OCR Revision

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.

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’.

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.

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.