Identifying Subgroups of the Modular Group via Permutation Representations on Cosets

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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!

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In what sense is $C$ known if you don't know what $H$ is (and hence don't know what the cosets of $H$ are)? Do you only know $C$ as an abstract group? Do you know its action on the cosets as an abstract group action?  – Qiaochu Yuan Aug 6 at 20:43

2 Answers

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 $H$.

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(I wanted to post this a comment on @Will Sawin's answer, but I don't have the rep yet. Also it turned out to be convenient to flesh it out a little longer than comment.)

Will's answer assumes that you know a fixed element of $G/H$ that you can use as your basepoint for the fundamental group, namely the identity coset $H$. If you don't know that, you can't recover $H$ exactly, you can only recover $H$ up to conjugacy. (The permutation actions of $G$ on $G/H$ and $G/H'$ are permutationally isomorphic iff $H$ and $H'$ are conjugate subgroups.)

For example, in the case of $PSL(2,\mathbb{Z})$, this means that subgroups-up-to-conjugacy of index $n$ are in bijective correspondence with pairs of permutations $(a,b)$ on $[n]$ such that $a^{2} = b^{3} = 1$ (picking some natural generators of $PSL(2,\mathbb{Z}) \cong \mathbb{Z}_2 * \mathbb{Z}_3$)

and $\langle a, b\rangle$ acts transitively on $[n]$, up to the relation $(a,b) \sim (a',b')$ if there is a permutation $\pi \in S_{n}$ such that $\pi a \pi^{-1} = a'$ and $\pi b \pi^{-1} = b'$ (permutational isomorphism).

Subgroups (not up to conjugacy, but on the nose) are classified by the same pairs, but where the $\pi$ that we are allowed to conjugate by must fix, say, $1 \in [n]$ (corresponding to the identity coset of $H$ in $G/H$).

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