Editing Congruence subgroup (section)

=== Principal congruence subgroups ===

If <math>n \geqslant 1</math> is an integer there is a homomorphism <math>\pi_n: \mathrm{SL}_2(\Z) \to \mathrm{SL}_2(\Z /n\Z)</math> induced by the reduction modulo <math>n</math> morphism {{tmath|1= \Z \to \Z / n\Z }}. The ''principal congruence subgroup of level <math>n</math>'' in <math>\Gamma = \mathrm{SL}_2(\Z)</math> is the kernel of {{tmath|1= \pi_n }}, and it is usually denoted {{tmath|1= \Gamma(n) }}. Explicitly it is described as follows:
: <math> \Gamma(n) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\Z) : a, d \equiv 1 \pmod n, \quad b, c \equiv 0 \pmod n \right\} </math>

This definition immediately implies that <math>\Gamma(n)</math> is a [[normal subgroup]] of finite [[Index of a subgroup|index]] in {{tmath|1= \Gamma }}. The [[strong approximation theorem]] (in this case an easy consequence of the [[Chinese remainder theorem]]) implies that <math>\pi_n</math> is surjective, so that the quotient <math>\Gamma /\Gamma(n)</math> is isomorphic to {{tmath|1= \mathrm{SL}_2(\Z/n\Z) }}. Computing the order of this finite group yields the following formula for the index:
: <math> [\Gamma : \Gamma(n)] = n^3 \cdot \prod_{p \mid n} \left( 1 - \frac 1 {p^2} \right) </math>
where the product is taken over all prime numbers dividing {{tmath|1= n }}.

If <math>n \geqslant 3</math> then the restriction of <math>\pi_n</math> to any finite subgroup of <math>\Gamma</math> is injective. This implies the following result: 
: ''If <math>n\geqslant 3</math> then the principal congruence subgroups <math>\Gamma(n)</math> are [[Torsion-free group|torsion-free]].''
The group <math>\Gamma(2)</math> contains <math>-\operatorname{Id}</math> and is not torsion-free. On the other hand, its image in <math>\operatorname{PSL}_2(\Z)</math> is torsion-free, and the quotient of the [[hyperbolic plane]] by this subgroup is a sphere with three cusps.