Groups: Orders of elements and groups

Understanding Orders of Elements in a Group

The order of an element within a group is the smallest positive integer, n, such that a^n equals the identity element (e). Here, ‘a’ is an element of the group and ‘^’ denotes the operation of the group repeatedly applied.
In symbols, this can be represented as: for a group (G, *), if an element a belongs to G, and there exists a positive integer n such that a^n = e, then n is known as the order of the element a.
If there is no such positive integer n, the element a is said to have infinite order.

The order of a group refers to the total number of elements in the group. It can be finite or infinite.
If a group G has a finite number of elements, we can say that G is a finite group and the order of G is the number of elements in G.
For example, if a group G has 4 elements, then we can say that the order of G is 4.
If a group has an infinite number of elements, then we can say that the group has infinite order.

Let’s consider a group (Z, +), which is the group of integers under addition. The order of the group is infinite as there are infinitely many integers.
The identity element in this group is 0. For any integer n, n+0=0+n=n, showing 0 is the identity element.
If we take an element a=2, we can see that there is no n such that 2n = 0. Thus, 2 has infinite order in this group.
Consider another group, the integers modulo 5 under addition, denoted as (Z5, +). The group has five elements [0,1,2,3,4] and hence the order of this group is 5.
In this group, the element 3 has order 5 because 5*3=15 and the remainder when 15 is divided by 5 is 0, which is the identity element.

Determining the order of an element or group can assist in the process of group classification, which is a fundamental challenge in abstract algebra.
The order properties have roles in various mathematical proofs and are a stepping stone to more advanced group properties such as cyclic groups and generators.
Understanding the order of elements and groups is significant in fields that utilise group theory, such as physics, cryptography, and theoretical computer science.