Section 9.1 Algebraic Structures
711
b. Show that the subset
a l = i
a 5 = (1, 2) + (3, 4)
a 2 = (1, 2, 3, 4)
a 6 = (1, 4) + (2, 3)
a 3 = (1, 3) + (2, 4)
a 7 = (2, 4)
a 4 = (1, 4, 3, 2)
a 8 = (1, 3)
forms a subgroup of the symmetric group S 4 .
15. Find the elements of the alternating group A 4 .
16. Let A = 5p, q, r6. Then 3A*, # 4 is the free monoid generated by A.
a. What is ppqrp # qprr ?
b. Let B = the set of all strings over A with an even number of q’s. Then B # A. Prove that 3B, # 4 is also
a monoid.
17. In each case, decide whether the given function is a homomorphism from the group on the left to the one
on the right. Are any of the homomorphisms also isomorphisms?
a. 3Z, + 4, 3Z, + 4; f (x) = 2
b. 3R, + 4, 3R, + 4; f (x) = 0 x 0
c. 3R*, # 4, 3R*, # 4 (where R* denotes the set of nonzero real numbers); f (x) = 0 x 0
18. In each case, decide whether the given function is a homomorphism from the group on the left to the one
on the right. Are any of the homomorphisms also isomorphisms?
a. 3R3x 4, + 4, 3R, + 4; f (a n x
n
+ a n−1 x
n−1
+ c + a 1 x + a 0 ) = a n + a n−1 + c + a 0
b. 3S 3 , + 4, 3Z 2 , + 2 4; f (a) = e
1 if a is an even permutation
0 if a is an odd permutation
c. 3Z × Z, + 4 where + denotes componentwise addition, 3Z, + 4; f (x, y) = x + 2y
19. In each case, decide whether the given groups are isomorphic. If they are, produce an isomorphism
function. If they are not, give a reason why they are not.
a. 3Z, + 4, 312Z, + 4 (where 12Z = 512z 0 z [ Z6)
b. 3Z 5 , + 5 4, 35Z, + 4
c. 35Z, + 4, 312Z, + 4
d. 3S 3 , + 4, 3Z 6 , + 6 4
20. In each case, decide whether the given groups are isomorphic. If they are, produce an isomorphism function. If they are not, give a reason why they are not.
a. 3 5a 1 x + a 0 0 a 1 , a 0 [ R6, + 4, 3C, + 4
b. 3Z 6 , + 6 4, 3S 6 , + 4
c. 3Z 2 , + 2 4, 3S 2 , + 4
d. 3Z
3
2 , + 2 4, 3Z 8 , + 8 4
21. Let M 2
0
(Z) be the set of all 2 × 2 matrices of the form
c
1 z
0 1
d
where z [ Z.
a. Show that 3M 2
0
(Z), # 4 is a group, where # denotes matrix multiplication.