Difference between revisions of "MA 453 Fall 2012 Walther Homework" - Rhea

Revision as of 02:22, 30 September 2012

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-Chapter 4:
-Problem 9
-From theorem 4.3 we know that "for each positive divisor k of n, the group <a> has exactly one subgroup of order k"  ==> <a^(n/k)>
-"For each positive divisor k of n, the set <n/k> is the unique subgroup of Zn of order k. Most importantly these are the only subgroups of Zn."
-Therefore if we follow the similiar examples on page 78 and 79 (example 5) we can see that for each divisor k of 20, the set <20/k> is the unique subgroup of Z20 of order k and are the only subgroups of Z20.
-Thus <1>, <2>, <4>, <5>, <10> and <20> are the subgroups of Z20.
-Similarily, from the example on page 78 we see that <a>, <a^2>  , <a^4>, <a^5>, <a^10>, <a^20> are the subgroups of G.
-The generators are just the divisors of 20 (for Z20) or are a^k with k being a divisor of 20 (for the set G).
-==> 1 , 2 , 4, 5, 10, 20
-==> a, a^2, a^4, a^5, a^10, a^20
 == Homework solutions ==
 * HW 1 is in [[http://www.math.purdue.edu/~walther/teach/453/hw/1.txt]]
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 * HW 4 is in [[http://www.math.purdue.edu/~walther/teach/453/hw/4.txt]]
 * HW 5 is in [[http://www.math.purdue.edu/~walther/teach/453/hw/5.txt]]
+[[Chapter 4: Problem 9]]