202 Groups 45. 30. Express U(165) as an internal direct product of proper subgroups 46. in four different ways. 31. Let R denote the group of all nonzero real numbers under multi plication. Let R denote the group of positive real numbers under multiplication. Prove that R* is the internal direct product of R+ and the subgroup {1, -1} 32. Prove that D, cannot be expressed as an internal direct product of two proper subgroups. 33. Let H and K be subgroups of a group G. If G = HK and g = hk. where h E H and k E K, is there any relationship among Igl, Ihl, 47. 48. 49. 4 50. and lkl? What if G = H X K? 51 34. In Z, let H = (5) and K = (7). Prove that Z = HK. Does Z = HX K {3a6 10c I a, b, c E Z} under multiplication and H = 35. Let G {3a6b12c I a, b, c E Z} under multiplication. Prove that G = (3) x (6) X (10), whereas H (3) x (6) X (12). 36. Determine all subgroups of R* (nonzero reals under multiplica- tion) of index 2 37. Let G be a finite group and let H be a normal subgroup of G. Prove that the order of the element gH in G/H must divide the order of g in G. 52 53 54 38. Let H bea normal subgroup of G and let a belong to G. If the ele- ment aH has order 3 in the group G/H and H =10, what are the possibilities for the order of a? 39. If H is a normal subgroup of a group G, prove tralizer of H in G, is a normal subgroup of G. 1t that C(H), the cen- 40. Let d be an isomorphism from a group G onto a group G. Prove that if H is a normal subgroup of G, then d(H) is a normal sub- group of G. 41. Show that Q, the group of rational numbers under addition, has no proper subgroup of finite index. 42. An element is called a square if it can be expressed in the form b for some b. Suppose that G is an Abelian group and H is a sub- group of G. If every element of H is a square and every element of GIH is a square, prove that every element of G is a square. Does your proof remain valid when "square" is replaced by "nth power, where n is any integer? 43. Show, by example, that in a factor group G/H it can happen that aH bH but lal lbl. 44. Observe from the table for A given in Table 5.1 on page 111 that the subgroup given in Example 9 of this chapter is the only sub- group of A, of order 4. Why does this imply that this subgroup must be normal in A? Generalize this to arbitrary finite groups. 4

Prove or disprove that H={ [ 1a01 ]|a } is a normal subgroup of the special linear group SL(2,).

let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.

Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .

Prove that each of the following subsets H of GL(2,C) is subgroup of the group GL(2,C), the general linear group of order 2 over C a. H={ [ 1001 ],[ 1001 ],[ 1001 ],[ 1001 ] } b. H={ [ 1001 ],[ i00i ],[ i00i ],[ 1001 ] }

Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .

Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.

Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.

11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.

13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by