Put $H=\operatorname{Stab}_F(x)$. For $g,h\in G$,
$$
\begin{aligned}
f_x(g)=f_x(h)
&\iff F(g,x)=F(h,x)\\
&\iff F(h^{-1}g,x)=x\\
&\iff h^{-1}g\in H\\
&\iff g\in hH.
\end{aligned}
$$
Thus $f_x$ is constant on every left <coset> of $H$ and takes different values on different cosets. Its oracle is therefore an oracle for the <hidden subgroup problem>, and the hidden subgroup is precisely the <stabilizer subgroup> $H$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 1 / a / ii / Solution

 Solution

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