Editing Modular group (section)

==Congruence subgroups==
{{Main|Congruence subgroup}}
Important [[subgroup]]s of the modular group {{math|Γ}}, called ''[[congruence subgroup]]s'', are given by imposing [[congruence relation]]s on the associated matrices.

There is a natural [[homomorphism]] {{math|SL(2, '''Z''') → SL(2, '''Z'''/''N'''''Z''')}} given by reducing the entries [[modulo operation|modulo]] {{math|''N''}}. This induces a homomorphism on the modular group {{math|PSL(2, '''Z''') → PSL(2, '''Z'''/''N'''''Z''')}}. The [[kernel (algebra)|kernel]] of this homomorphism is called the '''[[principal congruence subgroup]] of level {{math|''N''}}''', denoted {{math|Γ(''N'')}}. We have the following [[short exact sequence]]:

:<math>1\to\Gamma(N)\to\Gamma\to\operatorname{PSL}(2, \mathbb Z/N\mathbb Z) \to 1.</math>

Being the kernel of a homomorphism {{math|Γ(''N'')}} is a [[normal subgroup]] of the modular group {{math|Γ}}. The group {{math|Γ(''N'')}} is given as the set of all modular transformations

:<math>z\mapsto\frac{az+b}{cz+d}</math>

for which {{math|''a'' ≡ ''d'' ≡ ±1 (mod ''N'')}} and {{math|''b'' ≡ ''c'' ≡ 0 (mod ''N'')}}.

It is easy to show that the [[trace (linear algebra)|trace]] of a matrix representing an element of {{math|Γ(''N'')}} cannot be −1, 0, or 1, so these subgroups are [[torsion-free group]]s. (There are other torsion-free subgroups.)

The principal congruence subgroup of level 2, {{math|Γ(2)}}, is also called the '''modular group {{math|Λ}}'''. Since {{math|PSL(2, '''Z'''/2'''Z''')}} is isomorphic to {{math|[[symmetric group|''S''<sub>3</sub>]]}}, {{math|Λ}} is a subgroup of [[index of a subgroup|index]] 6. The group {{math|Λ}} consists of all modular transformations for which {{math|''a''}} and {{math|''d''}} are odd and {{math|''b''}} and {{math|''c''}} are even.

Another important family of congruence subgroups are the [[modular group Gamma0|modular group {{math|Γ<sub>0</sub>(''N'')}}]] defined as the set of all modular transformations for which {{math|''c'' ≡ 0 (mod ''N'')}}, or equivalently, as the subgroup whose matrices become [[upper triangular matrix|upper triangular]] upon reduction modulo {{math|''N''}}. Note that {{math|Γ(''N'')}} is a subgroup of {{math|Γ<sub>0</sub>(''N'')}}. The [[modular curve]]s associated with these groups are an aspect of [[monstrous moonshine]] – for a [[prime number]] {{math|''p''}}, the modular curve of the normalizer is [[genus (mathematics)|genus]] zero if and only if {{math|''p''}} divides the [[order (group theory)|order]] of the [[monster group]], or equivalently, if {{math|''p''}} is a [[supersingular prime (moonshine theory)|supersingular prime]].