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Let $\sigma, \tau \in S_{n} .$ Show that $\operatorname{sgn}(\tau \circ \sigma)=(\operatorname{sgn} \tau)(\operatorname{sgn} \sigma) .$ Thus, the product of two eve...

Question

Let $\sigma, \tau \in S_{n} .$ Show that $\operatorname{sgn}(\tau \circ \sigma)=(\operatorname{sgn} \tau)(\operatorname{sgn} \sigma) .$ Thus, the product of two even or two odd permutations is even, and the product of an odd and an even permutation is odd.

Let $\sigma, \tau \in S_{n} .$ Show that $\operatorname{sgn}(\tau \circ \sigma)=(\operatorname{sgn} \tau)(\operatorname{sgn} \sigma) .$ Thus, the product of two even or two odd permutations is even, and the product of an odd and an even permutation is odd.



Answers

Let $\sigma, \tau \in S_{n} .$ Show that $\operatorname{sgn}(\tau \circ \sigma)=(\operatorname{sgn} \tau)(\operatorname{sgn} \sigma) .$ Thus, the product of two even or two odd permutations is even, and the product of an odd and an even permutation is odd.


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