Suppose we have a known group G and an unknown subgroup H. The permutation representation of G on the cosets of H gives a permutation group C, which is known. Is it possible to identify the generators of H using this information? If so, how? If not, what further information is needed?


(For background, my G is the modular group PSL(2,Z); H is a not-necessarily-congruence subgroup of G. C is sometimes called a "cartographic group").


Many thanks!



Identifying Subgroups of the Modular Group via Permutation Representations on Cosets
Choose a set of generators for $G$. Draw the Cayley graph of the action of $G$ on $G/H$. Choose a set of generators for the fundamental group of the graph. For each generator of the fundamental group, multiply the generators of $G$ corresponding to the edges together in order. This will be a set of generators for $H$.


Proof: Each element of $H$ is a product of the generators of $G$ which, in the action of $G$ on $G/H$, preserves the identity coset. Thus it's a product of generators that forms a closed loop of the Cayley graph. Homotoping that loop to a composition of the generators of the fundamental group does not change the element it refers to. You now have an expression for each element of $H$ in terms of the supposed generators of $H$, so they are in fact the generators of $H$.


