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

Table of contents
- Solution ii

Solution

Put

H = Stab_{F} (x)

. For

g, h \in G

,

f_{x} (g) = f_{x} (h) ⟺ F (g, x) = F (h, x) ⟺ F (h^{- 1} g, x) = x ⟺ h^{- 1} g \in H ⟺ g \in h H .

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

.

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