# 202Groups45.30. Express U(165) as an internal direct product of proper subgroups46.in four differentways.31. Let R denote the group of all nonzero real numbers under multiplication. Let R denote the group of positive real numbers undermultiplication. 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 oftwo 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,47.48.49.450.and lkl? What if G = H X K?5134. 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 237. Let G be a finite group and let H be a normal subgroup of G. Provethat the order of the element gH in G/H must divide the orderof g in G.52535438. 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 thepossibilities for the order of a?39. If H is a normal subgroup of a group G, provetralizer of H in G, is a normal subgroup of G.1tthat C(H), the cen-40. Let d be an isomorphism from a group G onto a group G. Provethat 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 noproper subgroup of finite index.42. An element is called a square if it can be expressed in the form bfor 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 ofGIH is a square, prove that every element of G is a square. Doesyour 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 thataH bH but lal lbl.44. Observe from the table for A given in Table 5.1 on page 111 thatthe subgroup given in Example 9 of this chapter is the only sub-group of A, of order 4. Why does this imply that this subgroupmust be normal in A? Generalize this to arbitrary finite groups.4

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