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32. 33. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. multiplication. Prove that R;is the internal direct product of R+ and the subgroup [1, l}. Prove that D4 cannot be expressed as an internal direct product of two proper subgroups. Let H and K be subgroups of a group G. If G 2 HK and g = hk, where h E H and k E K, is there any relationship among Igt, lhl, and |k|?WhatifG = H X K? InZ,1etH = (5) andK= (7). vae thatZ= HK. DoesZ= H X K? Let G = {396510C | a, b, c E Z] under multiplication and H = {3\"6b12' l a, b, c E Z] under multiplication. Prove that G = (3) X (6) x (10), whereas H at (3) x (6) x (12). Determine all subgroups of R* (nonzero reals under multiplica tion) of index 2. Let G be a nite group and let H be a normal subgroup of G. Prove that the order of the element 315' in G/H must divide the order of g in G. Let H be a normal subgroup of G and let a belong to G. If the ele- ment aH has order 3 in the group G/H and IHI =10, what are the possibilities for the order of a? If H is a normal subgroup of a group G, prove that C(H), the cen tralizer of H in G, is a normal subgroup of G. Let ()5 be an isomorphism from a group G onto a group G. Prove that if H E a normal subgroup of G, then (NH) is a normal sub- group of G. Show that Q, the group of rational numbers under addition, has no proper subgroup of nite index. An element is called a square if it can be expressed in the form ()2 for some 22. 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 G/H 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? Show, by example, that in a factor group G/H it can happen that aH = 911 but lal Ibl. Observe from the table for A,1 given in Table 5.1 on page 111 that the subgroup given in Example 9 of this chapter is the only sub- group of A4 of order 4. Why does this imply that this subgroup