**The Order of an Element**

The order of an element in a group is the smallest positive power of the element which gives you the identity element. We discuss 3 examples:

elements of finite order in the real numbers, complex numbers, and a 2×2 rotation matrix.

**Basic Group Proof 1**

Let G be a group and show that if (ab)^2 = a^2b^2 for all a,b in G, then G is abelian.

**Basic Group Proof 2**

Let G be a group such that every element of G is equal to its own inverse. Show that G is abelian.

**Basic Group Proof 3**

Let G be a finite group of even order. Show that G has an element a (not equal to the identity) such that a^2=e.

We can understand easily