2. Joseph A. Gallian, Contemporary Abstract Algebra (4th Edition), Narosa Publishing House,

Answers to Chapter 7 Exercises Exercise 7. Find all of the left cosets of {1, 11} in U (30). Solution. We begin by listing the elements of the group and labeling the subgroup: U (30) = {1, 7, 11, 13, 17, 19, 23, 29}

H = {1, 11}.

So then by multiplication we have: 1H 7H 13H 19H

= {1, 11} = 11H = {7, 17} = 17H = {13, 23} = 23H = {19, 29} = 29H

Exercise 8. Suppose that a has order 15. Find all of the left cosets of ha5 i in hai. Solution. Note that ha5 i = {e, a5 , a10 }. So we have the following cosets by direct computation. I am listing only one representative name in each case: eha5 i = {e, a5 , a10 } aha5 i = {a, a6 , a11 } a2 ha5 i = {a2 , a7 , a12 } a3 ha5 i = {a3 , a8 , a13 } a4 ha5 i = {a4 , a9 , a14 } Exercise 15. Suppose that |G| = pq, where p and q are prime. Prove that every proper subgroup of G is cyclic. Proof. By Lagrange’s Theorem, we know that the order of any subgroups must divide pq, and so must be one of {1, p, q, pq}. Since proper subgroups must be smaller that the group, we are left with the trivial subgroup of order 1, and a prime for the order of any other subgroup. By Corollary 3 of Lagrange, we know that any group of prime order is cyclic. Exercise 19. Suppose G is a finite group of order n and m is relatively prime to n. If g ∈ G and g m = e, prove that g = e. Proof. For a contradiction, assume that g 6= e. Thus g m = e ⇒ |hgi| = k for some k > 1 that divides m. By Lagrange’s Theorem, we also know that k|n, hence gcd(m, n) = k. This contradicts the fact that n, m are relatively prime, so our assumption is false and g = e. Exercise 40. Let G be the group of rotations of a plane about a point P in the plane. Thinking of G as a group of permutations of the plane, describe the orbit of a point Q in the plane. Solution. Since rotations are isometries that preserve distance, each image of Q under this permutation is equidistant from P . Hence, the orbit of Q is a circle with center P and radius d(P, Q).