175 8 I External Direct Products 19. If r is a divisor of m and s is a divisor of n, find a subgroup of Z Z, that is isomorphic to Z, Z 20. Find a subgroup of Z, Zg that is isomorphic to Z, Z. 21. Let G and H be finite groups and (g, h) E GO H. State a necessary and sufficient condition for ((g, h)) = (g) {h). 18 22. Determine the number of elements of order 15 and the number of cyclic subgroups of order 15 in Z30 Z20 23. What is the order of any nonidentity element of Z Z Z? Generalize. of 24. Let m > 2 be an even integer and let n > 2 be an odd integer. Find a formula for the number of elements of order 2 in D D n- т 25. Let M be the group of all real 2 X 2 matrices under addition. Let N = ROROROR under componentwise addition. Prove that M and N are isomorphic. What is the corresponding theorem for the group of m Xn matrices under addition? 26. The group S, Z, is isomorphic to one of the following groups: Z12, Z Z, A, D Determine which one by elimination. 27. Let G be a group, and let H = {(g, g) | g E G}. Show that H is a subgroup of G G. (This subgroup is called the diagonal of GG.) When G is describe G G and H geometrically. 28. Find a subgroup of Z, Z, that is not of the form H K, where H is a subgroup of Z, and K is a subgroup of Z,. 29. Find all subgroups of order 3 in Z, Za. 30. Find all subgroups of order 4 in Z Z. 31. What is the largest order of any element in Z Z? 32. What is the order of the largest cyclic subgroup of Z, Z10 Z,? What is the order of the largest cyclic subgroup of Z, Z, n. O- 4' ral the set of real numbers under addition, his 4 4 this me ral- V 30 are phic 33. Find three cyclic subgroups of maximum possible order in Z ZtZ, of the form (a) (b) (c), where a E Z, bEZ and z ro- iso- 10 15 cE Z1S 34. How many elements of order 2 are in Z200000 Z4000000? Generalize. 35. Find a subgroup of Z Z200 that is isomorphic to Z, Z 36. Find a subgroup of Z12 Z Zs that has order 9. 37. Prove that R* R* is not isomorphic to C. (Compare this with iso- Exercise 15.) 38. Let neral Н- a, b E z L0 0

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175
8 I External Direct Products
19. If r is a divisor of m and s is a divisor of n, find a subgroup of Z
Z, that is isomorphic to Z, Z
20. Find a subgroup of Z, Zg that is isomorphic to Z, Z.
21. Let G and H be finite groups and (g, h) E GO H. State a necessary
and sufficient condition for ((g, h)) = (g) {h).
18
22. Determine the number of elements of order 15 and the number of
cyclic subgroups of order 15 in Z30 Z20
23. What is the order of any nonidentity element of Z Z Z?
Generalize.
of
24. Let m > 2 be an even integer and let n > 2 be an odd integer. Find
a formula for the number of elements of order 2 in D D
n-
т
25. Let M be the group of all real 2 X 2 matrices under addition. Let
N = ROROROR under componentwise addition. Prove that
M and N are isomorphic. What is the corresponding theorem for
the group of m Xn matrices under addition?
26. The group S, Z, is isomorphic to one of the following groups:
Z12, Z Z, A, D Determine which one by elimination.
27. Let G be a group, and let H = {(g, g) | g E G}. Show that H is a
subgroup of G G. (This subgroup is called the diagonal of
GG.) When G is
describe G G and H geometrically.
28. Find a subgroup of Z, Z, that is not of the form H K, where H
is a subgroup of Z, and K is a subgroup of Z,.
29. Find all subgroups of order 3 in Z, Za.
30. Find all subgroups of order 4 in Z Z.
31. What is the largest order of any element in Z Z?
32. What is the order of the largest cyclic subgroup of Z, Z10 Z,?
What is the order of the largest cyclic subgroup of Z, Z,
n.
O-
4'
ral
the set of real numbers under addition,
his
4
4
this
me
ral-
V
30
are
phic
33. Find three cyclic subgroups of maximum possible order in Z
ZtZ, of the form (a) (b) (c), where a E Z, bEZ and
z ro-
iso-
10
15
cE Z1S
34. How many elements of order 2 are in Z200000 Z4000000? Generalize.
35. Find a subgroup of Z Z200 that is isomorphic to Z, Z
36. Find a subgroup of Z12 Z Zs that has order 9.
37. Prove that R* R* is not isomorphic to C. (Compare this with
iso-
Exercise 15.)
38. Let
neral
Н-
a, b E z
L0 0
Transcribed Image Text:175 8 I External Direct Products 19. If r is a divisor of m and s is a divisor of n, find a subgroup of Z Z, that is isomorphic to Z, Z 20. Find a subgroup of Z, Zg that is isomorphic to Z, Z. 21. Let G and H be finite groups and (g, h) E GO H. State a necessary and sufficient condition for ((g, h)) = (g) {h). 18 22. Determine the number of elements of order 15 and the number of cyclic subgroups of order 15 in Z30 Z20 23. What is the order of any nonidentity element of Z Z Z? Generalize. of 24. Let m > 2 be an even integer and let n > 2 be an odd integer. Find a formula for the number of elements of order 2 in D D n- т 25. Let M be the group of all real 2 X 2 matrices under addition. Let N = ROROROR under componentwise addition. Prove that M and N are isomorphic. What is the corresponding theorem for the group of m Xn matrices under addition? 26. The group S, Z, is isomorphic to one of the following groups: Z12, Z Z, A, D Determine which one by elimination. 27. Let G be a group, and let H = {(g, g) | g E G}. Show that H is a subgroup of G G. (This subgroup is called the diagonal of GG.) When G is describe G G and H geometrically. 28. Find a subgroup of Z, Z, that is not of the form H K, where H is a subgroup of Z, and K is a subgroup of Z,. 29. Find all subgroups of order 3 in Z, Za. 30. Find all subgroups of order 4 in Z Z. 31. What is the largest order of any element in Z Z? 32. What is the order of the largest cyclic subgroup of Z, Z10 Z,? What is the order of the largest cyclic subgroup of Z, Z, n. O- 4' ral the set of real numbers under addition, his 4 4 this me ral- V 30 are phic 33. Find three cyclic subgroups of maximum possible order in Z ZtZ, of the form (a) (b) (c), where a E Z, bEZ and z ro- iso- 10 15 cE Z1S 34. How many elements of order 2 are in Z200000 Z4000000? Generalize. 35. Find a subgroup of Z Z200 that is isomorphic to Z, Z 36. Find a subgroup of Z12 Z Zs that has order 9. 37. Prove that R* R* is not isomorphic to C. (Compare this with iso- Exercise 15.) 38. Let neral Н- a, b E z L0 0
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