# ahw6 - Math 4124 Wednesday, March 23 Sixth Homework...

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Math 4124 Wednesday, March 23 Sixth Homework Solutions 1. 4.2.2 on page 121. List the elements of S 3 as 1, (1 2), (2 3), (1 3), (1 2 3), (1 3 2), and label these with the integers 1,2,3,4,5,6 respectively. Exhibit the image of each element of S 3 under the left regular representation of S 3 into S 6 . Let us do the cycle decomposition for (1 2). This element sends 1 to (1 2), (1 2) to (1 2)(1 2) = 1, (2 3) to (1 2)(2 3) = (1 2 3), (1 2 3) to (1 2)(1 2 3) = (2 3), (1 3) to (1 2)(1 3) = (1 3 2), (1 3 2) to (1 2)(1 3 2) = (1 3). It follows that the cycle decomposition of the image of (1 2) in S 6 is (1 2)(3 5)(4 6). The other cycle decompositions are done similarly. The ﬁnal answer is Element Cycle Decomposition 1 (1) (1 2) (1 2)(3 5)(4 6) (2 3) (1 3)(2 6)(4 5) (1 3) (1 4)(2 5)(3 6) (1 2 3) (1 5 6)(2 4 3) (1 3 2) (1 6 5)(2 3 4) 2. (a) The conjugacy classes of S 5 and their orders are given by the following table. Shape

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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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ahw6 - Math 4124 Wednesday, March 23 Sixth Homework...

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