Tట ee 5३/७ 334 Groups 240 39. Let G b 27. Let G be a finite group and let p be a prime. If p2> IGI, show that any subgroup of order p is normal in G. 28. Let G ZeZ and H = {(x, y) |x and y are even that H is a subgroup of G. Determine the order of G/H. To which mead integers). Show Н- Нa no familiar group is GIH isomorphic? 29. Let n be a positive integer. Prove that every element of order n in Q/Z is contained in (1/n + Z). 30. (1997 Putnam Competition) Let G be a group and let d: G-G be 40. Find a to Z € р 41. Recall every a function such that group s)(8)8) d(h,)(h,)(h,) subgr whenever g18,83 = e = h,h,hg. Prove that there exists an element a in G such that (x) = ap(x) is a homomorphism. 31. Prove that every homomorphism from Z Z into Z has the form (x, y)- ax + by, where a and b are integers. 32. Prove that every homomorphism from Z Z into Z Z has the form (x, y)-(ax + by, cx + dy), where a, b, c, and d are integers. 33. Prove that Q/Z is not isomorphic to a proper subgroup of itself. 34. Prove that for each positive integer n, the group Q/Z has exactly (n) elements of order n (d is the Euler phi function). 35. Show that any group with more than two elements has an automor- phism other than the identity mapping. 36. A proper subgroup H of a group G is called maximal if there is no subgroup no maximal subgroups. 37. Let G be the group of quaternions as given in Exercise 4 of te Supplementary Exercises for Chapters 1-4 and let H = (a). Determine whether G/H is isomorphic to Za or Z, Z. Is G/H iso- morphic to a subgroup of G? 38. Write the dihedral group Dg as (Ro, R45, Ro0, R135, R180, Rs, R0 R315, F, F2, F, F4, Fs, F6 F, Fs) and let N (Rp, R RI0 R7 Prove that N is normal in Ds. Given that F,N determine whether DglN is cyclic. K such that H CKC G. Prove that O under addition has 90* (F. F F F) 4*