Comment by vdelecroix for Hi,Beginner in Sage (I love it!) , I want to ask you this maybe naive question:I have a group $G$ defined by $\lbrace M \in SL_2(Z) \ | \ M = I + kB \ mod(N) \rbrace $, where $I$ is the identity, $N$ a given positive integer, $B$ is a given fixed matrix verifying $B^2 = 0$ and $k$ (not fixed) any integer in { 0,1,2,....,N-1 }. I want to explore conjugacy classes/subgroups of this group $G$ for some specific $B$ and $N$. Is this subgroup finitely generated for some $N$? subgroups of finite index? etc... Do I have a way to do that with SageMath?Remark thatIt is subgroup of a finitely generated group butv that do not imply that it is finitely generated$B^2 = 0$ gives that $I + kB = \Lambda^k \ with \ \Lambda = I + B$. An obvious subgroup is $ H = \lbrace \Lambda^k, \ k \in \mathbb Z \rbrace$Any advise or web pointer would be appreciatedThanks for your help

With this definition, it is the preimage of a subgroup under the reduction mod N: `SL(2,Z) -> SL(2,Z/NZ)`. Hence it is finite index in `SL(2,Z)` and in particular finitely generated. It is also congruence as it contains Gamma(N).