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ahw6 - Math 3124 Thursday October 20 Sixth Homework...

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Math 3124 Thursday, October 20 Sixth Homework Solutions 1. Problem 17.9. Assume that G is a group with a subgroup H such that | H | = 6, [ G : H ] > 4, and | G | < 50. What are the possibilities for | G | ? By Lagrange’s theorem and using the given hypotheses, | G | = 6 [ G : H ] < 50 and [ G : H ] > 4. Therefore the possibilities for [ G : H ] are 5,6,7,8 and hence the possibilities for | G | are 30, 36, 42 and 48. 2. Problem 17.17. Find all the subgroups of Z 36 . Also construct the subgroup lattice. Since Z 36 is a cyclic group of order 36, there will be exactly one subgroup for each positive integer dividing 36. The integers dividing 36 are 2 a 3 b where a , b = 0 , 1 , 2, so there will be 3 * 3 = 9 subgroups. The subgroup lattice will look like [ 0 ] [ 12 ] [ 18 ] [ 6 ] [ 2 ] [ 9 ] [ 4 ] [ 3 ] [ 1 ] 3. From ae = c , we see that neither a nor e is the identity, and from bd = c , we see that neither b nor d is the identity. Therefore c is the identity. Since c is the identity, ae = c yields ea = c , and bd = c yields db = c . We now have the following partial Cayley table a b c d e a a c b b c c a b c d e d c d e c e There are now two ways to complete the table. The first possibility is that
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