feb02 - Math 4124 Wednesday, February 2 February 2,...

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Math 4124 Wednesday, February 2 February 2, Ungraded Homework Exercise 2.3.1 on page 60 Find all subgroups of Z 45 = h x i , giving a generator for each. Describe the containments between these subgroups. The problem is equivalent to finding all the subgroups and containments between them for Z / 45 Z . For each positive integer n dividing 45, there is a unique subgroup of order n ; this subgroup is h 45 / n i , the cyclic group with generator 45 / n . Therefore the subgroups of Z / 45 Z are h 1 i , h 3 i , h 5 i , h 9 i , h 15 i , h 45 i . For the containments, we have h a i ⊆ h b i if and only if b ± ± a (this only works when a , b ± ± 45). Thus we have h 45 i is contained in h 1 i , h 3 i , h 5 i , h 9 i , h 15 i , h 45 i . h 15 i is contained in h 1 i , h 3 i , h 5 i , h 15 i . h 9 i is contained in h 1 i , h 3 i , h 9 i . h 5 i is contained in h 1 i , h 5 i . h 3 i is contained in h 1 i , h 3 i . h
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This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.

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