Adv Alegbra HW Solutions 48

Adv Alegbra HW Solutions 48 - 48(ii Give an example of two...

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48 (ii) Give an example of two permutations α and β for which (αβ) 2 ±= α 2 β 2 . Solution. There are many examples. One is α = ( 12 ) and β = ( 13 ) . Since both α and β are transpositions, α 2 = ( 1 ) = β 2 , and so α 2 β 2 = ( 1 ) . On the other hand, αβ = ( 132 ) , and (αβ) 2 = ( 132 ) 2 = ( 123 ) ±= ( 1 ) . 2.31 (i) Prove, for all i , that α S n moves i if and only if α 1 moves i . Solution. Since α is surjective, there is k with α k = i .I f k = i , then α i = i and α i = j ±= i , a contradiction; hence, k ±= i .Bu t α 1 i = k , and so α 1 moves i . The converse follows by replacing α by α 1 ,for 1 ) 1 = α . (ii) Prove that if α, β S n are disjoint and if αβ = ( 1 ) , then α = ( 1 ) and β = ( 1 ) . Solution. By (i), if α and β are disjoint, then α 1 and β are dis- joint: if β moves some i , then α 1 must f x i .B u t αβ = ( 1 ) implies α 1 = β , so that there can be no i moved by β . There- fore, β = ( 1 ) = α . 2.32 If n 2, prove that the number of even permutations in S n is 1 2 n ! . Solution. Let τ = ( 12 ) , and de f ne f : A n O n , where A n is the set of all even permutations in S n and O n is the set of all odd permutations, by f : α 7→ τα. If σ is even, then τσ is odd, so that the target of f is, indeed, O
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