Hidden Subgroup Zoo

Let <math>f</math> be function from a group <math>G</math> to a set <math>S</math> which is promised to be constant and distinct on different left cosets of an unknown subgroup <math>H</math>:

<math>f:H \rightarrow S</math> <math>f(g)=f(h)</math> iff <math>gH = gH</math>

The goal of the hidden subgroup problem is, by querying <math>f</math>, to identify the subgroup <math>H</math>.