abstract algebra: a study guide for beginners - Northern Illinois ...

34 CHAPTER 3. GROUPS
35.† Let G = Z × 10 × Z× 10 .
(a) If H = 〈(3, 3)〉 and K = 〈(3, 7)〉, list the elements of HK.
(b) If H = 〈(3, 3)〉 and K = 〈(1, 3)〉, list the elements of HK.
36. In G = Z × 16 , let H = 〈3〉 and K = 〈−1〉. Show that HK = G and H ∩ K = {1}.
37.† In G = Z × 15 , find subgroups H and K with |H| = 4, |K| = 2, HK = G, and
H ∩ K = {1}.
38. Let G be an abelian group with |G| = 60. Show that if a, b ∈ G with o(a) = 4 and
o(b) = 6, then a 2 = b 3 .
39. Let G be a group, with subgroup H. Show that K = {(x, x) ∈ G × G | x ∈ H} is a
subgroup of G × G.
40. For groups G1 and G2, find the center of G1 × G2 in terms of the centers Z(G1) and
Z(G2). (See Exercise 3.2.21 for the definition of the center of a group.)
�
1
41.† Find the orders of
3
� �
2 1
and
4 4
�
2
in GL2(Z5).
3
��
m
42. Let K =
0
� �
b
�
∈ GL2(Z5) �
1
� m, b ∈ Z5,
�
m �= 0 .
(a) Show that K is a subgroup of GL2(Z5) with |K| = 20.
†(b) Show, by finding the order of each element in K, that K has elements of order 2
and 5, but no element of order 10.
43. Let H be the subgroup of upper triangular matrices in GL2(Z3).
(a) Show that with |H| = 12.
� �
−1 1
(b) Let A =
and B =
0 −1
� −1 0
0 1
�
. Show that o(A) = 6, o(B) = 2.
(c) Show that BA = A−1B. ⎧⎡
⎨ 1
44. Let H = ⎣ 0
⎩
0
a
1
0
⎤�
⎫
b �
� ⎬
c ⎦�
� a, b, c ∈ Z2
1 � ⎭ . Show that H is a subgroup of GL3(Z2). Is H
an abelian group?
⎡
0 0 1
⎤
0
⎢
45.† Find the cyclic subgroup of GL4(Z2) generated by ⎢ 0
⎣ 0
0
1
0
1
1 ⎥
0 ⎦
1 0 1 0
.
46. Let F be a field. Recall that a matrix A ∈ GLn(F ) is called orthogonal if A is
invertible and A −1 = A T , where A T denotes the transpose of A.
(a) Show that the set of orthogonal matrices is a subgroup of GLn(F ).