Difference between revisions of "Chapter 4: Chapter 4: Problem 9" - Rhea

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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

Latest revision as of 03:23, 30 September 2012