Math 444
Midterm Solutions
1. On the set S of all real numbers except 1, dene a b = ab a b + 2. Assuming that
this is a binary operation on S (it is), show that (S, ) is a group.
Solution: We have to check G1 , G2 and G3 .
G1 (associativity): We compute a
Math 444
100 Points
Name:
Midterm
Show Your Work
1. On the set S of all real numbers except 1, dene a b = ab a b + 2. Assuming that
this is a binary operation on S (it is), show that (S, ) is a group.
15
2. Let F be the set of all real-valued functions of
Math 444
Homework #8
(3.3) #2. Find the order of the factor group (Z4 Z12 )/( 2 2 )
Solution: First, the order of Z4 Z12 is 4 12 = 48. Also, the order of 2 Z4 is 2 (since
2 = cfw_0, 2). The order of 2 Z12 is 6 since 2 = cfw_0, 2, 4, 6, 8, 10. Hence, the o
Math 444
Homework #7
(3.1) #1. Determine if the map is a homomorphism. Let : Z R under addition
be given by (n) = n.
Solution: This is a homomorphism since for integers n and m, the sum n + m is the same
whether we consider them as integers or real number
Math 444
Homework #4
(1.4) #1. Find the quotient and remainder, according to the division algorithm,
when n is divided by m.
n = 42, m = 9
Solution: The multiple of 9 that is just to the left of 42 on the number line is 36 = 4 9,
so q = 4. The remainder i
Math 444
Homework #3
(1.3) #12. Determine whether the given set of invertible n n matrices with real
number entries is a subgroup of GL(n, R).
The set of all n n matrices A such that AT A = In .
Solution: Call H = cfw_A GL(n, R) | AT A = I . We need to ch
Math 444
Homework #2
(1.1) #2. With as dened in Table 1.4, compute (a b) c and a (b c). Can you
say on the basis of this computation whether is associative?
Solution: Well, (a b) c = b c = a and a (b c) = a a = a. This is consistent with
being associativ
Math 444
Homework #1
(0.1) #10. This exercise illustrates experimentation leading to a conjecture. Draw
a fairly large circle, and mark one point on the circle. There is just one,
undivided region enclosed by the circle; write the number 1 down below
the