What is the number of even permutations in S6? If a permutation have two or more cyclic decomposition , then they are either an even permutation or odd permutation. I am having promblem while listing the permutations. I am studying group theorey all by myself. Do tell me how to calculate the even and odd permutations in a symmetric group. Thankyou
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$\begingroup$ Are you familiar with the sign of a permutation giving a homomorphism? $\endgroup$– Tobias KildetoftOct 10, 2018 at 8:51
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$\begingroup$ No sir I am not $\endgroup$– Akshie DhimanOct 10, 2018 at 9:00
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$\begingroup$ Ok, another way then: Consider the map $S_6\to S_6$ given by $\sigma\mapsto (12)\sigma$. Do you see that this is a bijective map? What does it do to even permutations? $\endgroup$– Tobias KildetoftOct 10, 2018 at 9:02
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The sign function ${\rm sgn}:S_n\rightarrow \{\pm 1\}$ assigning to each permuation $\pi$ the sign ${\rm sgn}(\pi)$ is for $n\geq 2$ a surjective homomorphism. As it is a group homomorphism, each image is taken on equally often. Thus $A_n = \{\pi\mid{\rm sgn}(\pi)=1\}$ has cardinality $n!/2$.
