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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
=
a
(
1 2 3 4 5
)
A
be a subgroup generated by a 5cycle. Is it a normal
subgroup?
Solution:
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This homework help was uploaded on 02/01/2008 for the course MATH 150A taught by Professor Kuperberg during the Spring '03 term at UC Davis.
 Spring '03
 Kuperberg
 Algebra

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