Let G be a finite group. Then the number of elements in G is called the **order of the group** G and is denoted by **O(G)** or **|G|**

**Definition:**

Let *G* be group and *a **ϵ** G*. IF there exists a positive integer *n* such that *a ^{n} = e*, then the least such positive integer

*n*is called the order of

*a*.

If no such positive integer exists then we say that *a* is of infinite order.

**Example:**

G = { 1, -1, i, -I}

O(G) = 4

Let G be a group and a, b ϵ G with O(a) = 5, a^3b = ba^3 .Prove that G is an abelian group.