Adv Alegbra HW Solutions 45

Adv Alegbra HW Solutions 45 - 45 (vii) Every transposition...

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45 (vii) Every transposition is an even permutation. Solution. False. (viii) If a permutation α is a product of 3 transpositions, then it cannot be a product of 4 transpositions. Solution. True. (ix) If a permutation α is a product of 3 transpositions, then it cannot be a product of 5 transpositions. Solution. False. (x) If σασ 1 = ωαω 1 , then σ = ω . Solution. False. 2.22 Find sgn (α) and α 1 , where α = µ 123456789 987654321 . Solution. In cycle notation, α = ( 19 )( 28 )( 37 )( 46 ) . Thus, α is even, being the product of four transpositions. Moreover, being a product of disjoint transpositions, α = α 1 . 2.23 If σ S n f xes some j , where 1 j n (that is, σ( j ) = j ), de f ne σ 0 S X , where X ={ 1 ,..., b j n } ,by σ 0 ( i ) = i ) for all i ±= j . Prove that sgn 0 ) = sgn (σ). Solution. One of the cycles in the complete factorization of σ is the 1- cycle ( j ) . Hence, if there are t cycles in the complete factorization of σ , then there are
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