Modern Algebra : Some Properties and Proving involving Group

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.