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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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
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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

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If H is a nromal subgroup of G and a E G, then G laH x |H

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