# det - MORE ON DETERMINANTS Math 375 Preliminaries on...

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MORE ON DETERMINANTS Math 375 Preliminaries on permutations Let Z n = { 1 , 2 , 3 , . . . , n } . A function σ : Z n Z n which is bijective (i.e. one-to-one and onto 1 ) is called a permutation of the numbers 1 , 2 , 3 , . . . , n (or a permutation on n “letters”). We denote the set of all permutations on n letters by S n . As an exercise prove that the cardinality of S n is n !. A transposition τ is a permutation in S n which interchanges two numbers j, k , j = k and leaves the others fixed. Theorem 1. Let n 2 . Each permutation σ S n is a composition of transpositions. Proof by induction. Clearly the assertion is true for n = 2. We give the induction step, now assuming that the hypothesis for n - 1 is true. There is a k Z n so that σ ( k ) = n since σ is onto. Let τ be the transposition which flips k and n . Then τ ( n ) = k and thus σ ( τ ( n )) = σ ( k ) = n so that σ τ satisfies σ τ ( n ) = n . Thus σ τ maps Z n - 1 to itself. Therefore, by the induction hypothesis σ τ is a composition of transpositions. σ τ = τ 1 τ 2 ◦ · · · ◦ τ N . Since τ τ = id (identity map) we have σ = σ τ

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