= Inversion of an odd cycle in an alternating group
{title2=$g\sim_{A_n}g^{-1}\Longleftrightarrow n\equiv1\pmod4\quad(n\text{ odd})$}
For an odd $n$-cycle $g$, an inverting permutation fixes one point and swaps $(n-1)/2$ pairs, so its sign is $(-1)^{(n-1)/2}$. Its other inverting permutations differ by elements of $C_{S_n}(g)=\langle g\rangle$, all even. Thus inversion is possible inside $A_n$ exactly when that sign is positive. In particular $A_n$ is not an <ambivalent group> when $n\equiv3\pmod4$.
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