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Math 150a: Modern Algebra Homework 2 This problem set is due Wednesday, October 10. Starred problems may be harder and will be counted as extra credit. This includes both my starred problems and those in the book. Do problems 2.2.16(a,b), 2.2.17, 2.2.20, 2.M.3, 6.6.2, 6.6.4, and 6.6.10(a), in addition to the following: GK1. (I think that this is the same as 1.4.7(a).) Make a 3 × 3 matrix whose entries are 9 different variables, and compute its determinant by expanding minors. Compare with the permutation formula from class. *GK2. Let a and b be permutations in S n whose non-trivial cycles intersect in exactly one element. (For example, a = ( 1 3 5 )( 2 4 ) and b = ( 5 6 8 ) in S 8 .) Show that the commutator
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Unformatted text preview: aba-1 b-1 is a 3-cycle. GK3. The “15 puzzle” is a puzzle with 15 sliding tiles in a 4 × 4 tray that you may have seen from time to time. In its solved position, the puzzle looks like this: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 The puzzle lets you switch the empty square with any tile that is next to it. The usual object of the puzzle is to put the puzzle in its solved position. Show that, starting from the solved position, it is not possible to switch the 14 and the 15, and leave all of the other tiles unchanged. (Hint: You should use odd and even permutations, but there is more to it than that.)...
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This homework help was uploaded on 02/01/2008 for the course MATH 150A taught by Professor Kuperberg during the Spring '03 term at UC Davis.

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