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hw4 - Math 113 Homework 4 due 9/30/9 at 2:10 PM 1 Fraleigh...

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Math 113 Homework # 4, due 9/30/9 at 2:10 PM 1. Fraleigh section 8, problem 21; section 9, problem 23. 2. For f S n , define the support of f to be supp( f ) = { i ∈ { 1 , . . . , n } | f ( i ) 6 = i } . (a) Show that if f, g S n and supp( f ) supp( g ) = then fg = gf . (b) Extra credit: Show that if f, g S n and | supp( f ) supp( g ) | = 1, then fgf - 1 g - 1 is a 3-cycle. (This is used in solving Rubik’s cube.) 3. (a) Show that if μ = ( x 1 x 2 · · · x k ) S n is a k -cycle and σ S n is any permutation then σμσ - 1 is the k -cycle σμσ - 1 = ( σ ( x 1 ) σ ( x 2 ) · · · σ ( x k )) . (b) Using the above, can you guess a necessary and sufficient condition for two permutations in S n to be conjugate to each other? 4. Show that every even permutation in A n can be expressed as a product of (not necessarily disjoint) 3-cycles. Hint : use induction on n , imitat- ing one of the proofs given in class that every permutation in S n is a product of transpositions. 5. Let G be the symmetry group of a cube (with no reflections allowed). Show that G S 4 . Hint: a cube has four “diagonals” which connect
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