# Tటee5३/७334Groups24039. Let G b27. Let G be a finite group and let p be a prime. If p2> IGI, show thatany subgroup of order p is normal in G.28. Let G ZeZ and H = {(x, y) |x and y are eventhat H is a subgroup of G. Determine the order of G/H. To whichmeadintegers). ShowН-Нa nofamiliar group is GIH isomorphic?29. Let n be a positive integer. Prove that every element of order n inQ/Z is contained in (1/n + Z).30. (1997 Putnam Competition) Let G be a group and let d: G-G be40. Find ato Z €р41. Recalleverya function such thatgroups)(8)8) d(h,)(h,)(h,)subgrwhenever g18,83 = e = h,h,hg. Prove that there exists an element ain 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 theform (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 nosubgroupno maximal subgroups.37. Let G be the group of quaternions as given in Exercise 4 of teSupplementary 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, R0R315, F, F2, F, F4, Fs, F6 F, Fs) and let N (Rp, R RI0 R7Prove that N is normal in Ds. Given that F,Ndetermine whether DglN is cyclic.K such that H CKC G. Prove that O under addition has90*(F. F F F)4*

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