# 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.

## Understanding Orders of Groups

• 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.

## Examples of Orders in Groups

• 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.

## Significance of Orders in Group Theory

• 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.