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# hw4 - Math 113 Homework 4 due at the beginning of section...

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Unformatted text preview: Math 113 Homework 4, due 2/16/2012 at the beginning of section Policy: if you worked with other people on this assignment, write their names on the front of your homework. Remember that you must write up your solutions independently. 1. Fraleigh, section 8, problem 21; section 9, problem 23. 2. (a) Exhibit an element of S 9 which has order 20. (b) Prove that S 9 has no element of order 18. (c) A perfect shuffle of a deck of cards can be represented by the following permutation f S 52 : f ( x ) = braceleftBigg 2 x 1 , if x { 1 , . . . , 26 } , 2( x 26) , if x { 27 , . . . , 52 } . Show that, if one performs 8 perfect shuffles of a deck of cards, then this returns the cards to their original position. (Suggestion: first decompose f into a product of disjoint cycles.) 3. (a) Show that if = ( x 1 x 2 x k ) S n is a k-cycle and S n is any permutation, then - 1 is the k-cycle - 1 = ( ( x 1 ) ( x 2 ) ( x k ))....
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