Unformatted text preview: group ( Z / 36 Z ) × : 1,1, 5, 13,13, 17. Answer: 1,2,6,3,6,2. I won’t give explanations for all the answers here, just for 5. We have modulo 36, 5 1 = 5, 5 2 = 25, 5 3 = 17, 5 4 = 13, 5 5 = 29, 5 6 = 1. Therefore the least positive power of 5 which is the identity is 6, consequently the order of 5 is 6. 3. 1.1.25 on page 22 Prove that if G is a group and x 2 = 1 for all x ∈ G , then G is abelian. Let x , y ∈ G . Then x 2 = y 2 = ( xy ) 2 = 1. Therefore x 2 y 2 = 1 = ( xy ) 2 = xyxy . Multiplying by x1 on the left and y1 on the right, we obtain xy = yx as required....
View
Full Document
 Spring '08
 Staff
 Algebra, Multiplication, Prime number, nonzero rational number

Click to edit the document details