4023f10ps6

4023f10ps6 - 2 S 4 . (a) = (b) = (c) = id 7. Factor each of...

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Homework #6 Due: October 18, 2010 1. Find all generators of the cyclic group G = ( g ) if: (a) o ( g ) = 5 (b) o ( g ) = 10 (c) j G j = 16 (d) j G j = 20 2. In each case determine whether G is cyclic. Recall that Z / n is the group of invertible congruence classes modulo n under multiplication of congruence classes modulo n . (a) G = Z / 7 (b) G = Z / 12 (c) G = Z / 16 (d) G = Z / 11 3. Let o ( g ) = 20 in a group G . Compute: (a) o ( g 2 ) (b) o ( g 8 ) (c) o ( g 5 ) (d) o ( g 3 ) 4. (a) Suppose that G is a group of order 96 and that H is a subgroup of G such that there are 6 left cosets of H in G . What is j H j ? (b) Suppose that K and L are subgroups of a group M and assume that the following data are given: j K j = 9, j L j = 12, j M j < 100. What are the possible values of j M j ? 5. In this exercise G will denote the group Z / 60 of multiplicatively invertible congruence classes of Z 60 , with the group operation being multiplication of congruence classes modulo 60. (a) Verify that H = f [1] 60 ; [13] 60 ; [37] 60 ; [49] 60 g is a subgroup of G . (b) How many distinct left cosets does H have in G ? 6. Let ± = ( 1 2 3 4 2 4 1 3 ) and ¿ = ( 1 2 3 4 3 4 1 2 ). In each case, solve the given group equation for
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Unformatted text preview: 2 S 4 . (a) = (b) = (c) = id 7. Factor each of the following permutations into disjoint cycles. Using your answer express the inverse as a product of disjoint cycles. (a) 1 2 3 4 5 6 7 8 9 4 7 9 8 2 1 6 3 5 (b) 1 2 3 4 5 6 7 8 9 6 4 8 9 3 1 7 5 2 (c) 1 3 2 5 7 3 8 5 (d) 3 5 7 2 3 4 7 6 3 5 7 1 8. (a) How many permutations in S 5 x 1? (b) How many permutations in S 5 x both 1 and 3? 9. Let G = S 4 be the group of permutations of the set f 1 ; 2 ; 3 ; 4 g . For the purposes of this exercise it will probably be most convenient to write all elements of G in the disjoint cycle format. (a) Give a list of all of the distinct cyclic subgroups of G of order 4. (b) Given an example of a subgroup H of G of order 4 that is not cyclic. Math 4023 1...
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