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1. Let H = { (1), (12)(34), (13)(24), (14)(23) }. Find the left cosets of H in A (see Table 5.1 on page 1 11). 2. Let H be as in Exercise 1. How many left cosets of H in S, are there? (Determine this without listing them.) 3. Let H = {0, +3, +6, +9, . . .}. Find all the left cosets of H in Z. 4. Rewrite the condition a -1b C H given in property 5 of the lemma on page 145 in additive notation. Assume that the group is Abelian. 5. Let H be as in Exercise 3. Use Exercise 4 to decide whether or not the following cosets of H are the same. a. 11 + H and 17 + H b. - 1 + H and 5 + H c. 7 + H and 23 + H 6. Let n be a positive integer. Let H = {0, +n, +2n, +3n, . . .}. Find all left cosets of H in Z. How many are there? 7. Find all of the left cosets of {1, 11 } in U(30). 8. Suppose that a has order 15. Find all of the left cosets of (a ) in (a). 9. Let lal = 30. How many left cosets of (at) in (a) are there? List them. 10. Give an example of a group G and subgroups H and K such that HK = {h E H, k E K} is not a subgroup of G. 11. If H and K are subgroups of G and g belongs to G, show that g(H n K) = gH Q gK. 12. Let a and b be nonidentity elements of different orders in a group G of order 155. Prove that the only subgroup of G that contains a and b is G itself. 13. Let H be a subgroup of R*, the group of nonzero real numbers un- der multiplication. If R+ C H C R*, prove that H = R+ or H = R". 14. Let C" be the group of nonzero complex numbers under multiplica- tion and let H = (a + bi E C* la + b2 = 1}. Give a geometric de