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

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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 = x , 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 45 / n , the cyclic group with generator 45 / n . Therefore the subgroups of Z / 45 Z are 1 , 3 , 5 , 9 , 15 , 45 . For the containments, we have a b if and only if b a (this only works when a , b 45). Thus we have 45 is contained in 1 , 3 , 5 , 9 , 15 , 45 . 15 is contained in 1 , 3 , 5 , 15 . 9 is contained in 1 , 3 , 9 . 5 is contained in 1 , 5 . 3 is contained in 1 , 3 . 1 is contained in 1 . For the cyclic subgroup of order 45 with generator x , replace n with x n everywhere. Thus, for example, the subgroups of x are x , x 3 , x 5 , x 9 , x 15 , x 45 . The last one is of course 1 (the subgroup consisting of just the identity). Exercise 2.3.3 on page 60 Find all generators for Z / 48 Z . Z / 48 Z = { 0 , 1 ,..., 47 } . Also a is a generator of Z / 48 Z if and only if | a | = 48, and | a | = 48 / ( 48 , a ) . Thus a is a generator of Z / 48 Z if and only if ( a , 48 ) = 1. Therefore the generators of Z / 48 Z are 1 5 7 11 13 17 19 23 25 29 31
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