Algebra

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Kuperberg (11/14/07) Math 150a: Modern Algebra Second Midterm Solutions I decided to post students’ solutions that I liked for these questions. This way you can see real examples of good work. 1. In the additive group , what is 10 + 25 in set arithmetic? Solution: Solution by Valerie Rothfuss: This solution is not quite as rigorous as it could be, but I had trouble finding a complete explanation. One explanation is as follows: 5 is the union of 10 (the numbers that end in 0) and 10 + 5 (the numbers that end in 5). Then 10 = 10 + 0 and 10 + 5 = 10 + 25. So that way, 5 is a subset of 10 + 25 . But it’s also a superset, because every number of the form 10 a + 25 b is divisible by 5. If you quoted the theorem that a + b = gcd ( a , b ) , that’s okay as a start, but I didn’t do this theorem directly and it deserves explanation. 2. In the group S 5 , let H = ( ( 1 2 3 4 5 ) ) be a subgroup generated by a 5-cycle. Is it a normal subgroup? Solution: From Alex Aquila:
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Short and sweet.
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