View source for Chapter 3, Problem 9 MA453Fall2008walther - Rhea

The problem says show that U(20) does not equal <k> for any k in U(20). [Hence U(20) is not cyclic.] 
I was trying to understand Example 1 from the Chapter 3 in the text book. where it discusses U(15).
I am completely confused about what it is talking about:

U(15) = {1, 2, 4, 7, 8, 11, 13, 14 }
Then it goes on to say to find the order of element 7, so |7| = 4 
<math>7^1 = 7   7^2 =4   7^3 = 13    7^4 = 1</math>