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Suppose $H$ is a subgroup of a permutation group $S_n$ (with $n\geq 3$) which is not contained in $A_n$. Then as explained here the number of odd and even permutations of $H$ are equal. Does that mean order of such a subgroup is always even?

In other words, what can we say about the number of odd permutations in an odd-ordered subgroup?

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N. F. Taussig
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Mike V.D.C.
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  • I’ve added a much simpler proof to the linked question. – Robert Shore May 21 '23 at 07:53
  • @ThePhoenix My main question is about the odd-ordered subgroup. If I were to discard all possibilities, then such a subgroup is contained in $A_n$; hence, there is no permutation of odd order. – Mike V.D.C. May 21 '23 at 17:19

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