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# HW2d - Math 103A Homework Solutions HW2 Jacek Nowacki...

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Math 103A Homework Solutions HW2 Jacek Nowacki January 28, 2007 Chapter 2 Problem 8. Show that the set { 5 , 15 , 25 , 35 } is a group under multiplication modulo 40. What is the identity element of this group? Can you see any relationship between this group and U (8) ? Proof. In order to make our calculations simpler we will notice that: 25 = - 15 mod (40) and 35 = - 5 mod (40). This will give us the following cayley table 5 15 - 15 - 5 5 - 15 - 5 5 15 15 - 5 - 15 15 5 - 15 5 15 - 15 - 5 - 5 15 5 - 5 - 15 Notice the ”symmetry” in the above table. We see that the set is closed under multiplication. Also, it is quite clear that - 15, or 25 is the identity element, and each element is its own inverse. Associativity follows from the properties of integers. Hence, the set is a group. The group is closely related to U (8) = { 1 , 3 , 5 , 7 } via 25 · { 1 , 3 , 5 , 7 } = { 25 , 35 , 5 , 15 } mod (40). Problem 12. For any integer n > 2, show that there are at least two ele- ments in U ( n ) that satisfy x 2 = 1. Proof. We all know the solution of the above equaiton over the integers, namely x = 1 , - 1. Let’s see if we can translate it to U ( n ). First, we note that since n > 2, 1 = - 1 mod ( n ). Indeed, we will get two distinct elements 1

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as desired. Next, we see that 1 U ( n ) as gcd (1 , n ) = 1. To handle - 1, we notice that - 1 = n - 1 mod ( n ). So, all we have to do now is show that the gcd ( n, n - 1) = 1. But, (1) · n + ( - 1) · ( n - 1) = 1. Hence, gcd ( n, n - 1) = 1 (we’re using the fact that 1 is the smallest possitive integer). Problem 14. Let G be a group with the following property: Whenever a , b , and c belong to G and ab = ca , then b = c . Prove that G is Abelian.
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HW2d - Math 103A Homework Solutions HW2 Jacek Nowacki...

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