For example lets say you have a p group where every element of the group eventually gets to 1 if you multiply it by itself enough times. This is called a "p to the n" if n is the number of times you are doing this.
Did you know that every subgroup of this p^n that is all the p^n-1, p^n-2, etc are all normal? (normal means gx=xg) I did not know that until I read about p groups in Russian. I also I wanted to share  a way to show how this is true. We will call all the elements of the big group g (i.e. g1 g2 g3 etc) and all the elements of the small group x (i.e. x1 x2 x3  etc.) So what you have is a long string of gx's all lined up with xg's on the opposite side of your equation.
and they all cancel because if you take that many gs with that many x's you get e*e=e*e. What is left on the left side is just one g. and what is left on the right side? Also just one g.