Editing Congruence subgroup (section)

=== Definition of a congruence subgroup ===

A subgroup <math>H</math> in <math>\Gamma = \mathrm{SL}_2(\Z)</math>  is called a ''congruence subgroup'' if there exists <math>n \geqslant 1</math> such that <math>H</math> contains the principal congruence subgroup {{tmath|1= \Gamma(n) }}. The ''level'' <math>l</math> of <math>H</math> is then the smallest such {{tmath|1= n }}.

From this definition it follows that:
* Congruence subgroups are of finite index in {{tmath|1= \Gamma }}; 
* The congruence subgroups of level <math>\ell</math> are in one-to-one correspondence with the subgroups of {{tmath|1= \operatorname{SL}_2(\Z/\ell\Z ) }}.