# Tagged Questions

**4**

votes

**4**answers

501 views

### Are there noncongruence subgroups (of finite index) of the modular group generated only by 2 or 3 elements?

By the modular group I either mean $SL(2,\mathbb{Z})$, or $PSL(2,\mathbb{Z})$.
Where can I find examples of these?
Another question - is there a good (ideally analytical, but possibly ...

**3**

votes

**1**answer

184 views

### Simplifying presentations of modular subgroups

I've been using the Reidemeister-Schreier process (detailed in e.g. Holt et al. - Handbook of Computational Group Theory) to find the presentations of various modular subgroups. For example, this ...

**7**

votes

**3**answers

601 views

### Are congruence subgroups of the modular group finitely presented?

Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma_{0}\left(n\right)$, $\Gamma_{1}\left(n\right)$ etc.) ...

**1**

vote

**2**answers

210 views

### Identifying Subgroups of the Modular Group via Permutation Representations on Cosets

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 ...

**8**

votes

**1**answer

544 views

### Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...

**10**

votes

**3**answers

748 views

### Congruence subgroups as abstract groups

This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup
$$
\pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & ...

**8**

votes

**3**answers

512 views

### Congruence Subgroups as Open Subgroups of the Modular Group Under the Right Topology

It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topological group is open iff ...