Math 360 ALGEBRA HOMEWORK 8 (EXTRA) SOLUTIONS
Chapter 9 #4. Is H =
No. For example, let A =
Then ABA1 =
a b
a normal subgroup of GL(2, R)?
0 d
0 1
1 1
,B=
H.
1 0
0 1
1 0
which is not in H, so H is not normal in G.
1 1
Chapter 9 #10. Prove that a factor g
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Math 360 ALGEBRA HOMEWORK 10 SOLUTIONS
Problem 1. Let D be an integral domain. If n is the characteristic of D then n1 = 0.
If n = pq for primes p and q, then (pq)1 = 0.
Since (pq)1 = (p1)(q1) (why?), we have (p1)(q1) = 0. Because D has no zero divisors e
Math 360 ALGEBRA HOMEWORK 7 SOLUTIONS
a 2b
Chapter 6 #24. G = cfw_a + b 2 | a, b Q H = cfw_
b a
| a, b Q.
a 2b
Dene : G H by (a + b 2) =
.
b a
1. is 1-1 and onto. (Why?)
2. (a1 + b1 2) + (a2 + b2 2) = (a1 + b1 2) + (a2 + b2 2) (show!)
By 1 and 2 above G a
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