Answer to Question #178275 in Abstract Algebra for 123

Question #178275

Prove that for every subgroup H of Sn , either all of the permutations in H are even, or exactly half are even.



1
Expert's answer
2021-04-15T07:52:56-0400

Suppose that there is at least one odd permutation τ\tau (if there's no odd permutations, we conclude that all permutations are even). Then, for every even permutation αH\alpha \in H we obtain distinct odd permutations ατH\alpha \cdot \tau \in H (as if α1τ=α2τ\alpha_1 \cdot \tau = \alpha_2\cdot \tau, then by multiplying by τ1\tau^{-1} on the right we get that α1=α2\alpha_1=\alpha_2) and therefore we conclude that there is at least as many odd permutations as even permutations. However, for every odd permutation β\beta (including the case β=τ\beta=\tau) we can apply the same argument: βτ\beta \cdot \tau are distinct even permutation. Therefore there is at least as many even permutations odd permutations. By double inequality we conclude that there the same quantity of even and odd permutations, i.e., exactly half of permutations are even.


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