Modular Group - Congruence Subgroups

Important subgroups of the modular group Γ, called congruence subgroups, are given by imposing congruence relations on the associated matrices.

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

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Being the kernel of a homomorphism Γ(N) is a normal subgroup of the modular group Γ. The group Γ(N) is given as the set of all modular transformations

for which a ≡ d ≡ ±1 (mod N) and b ≡ c ≡ 0 (mod N).

The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ. Since PSL(2,Z/2Z) is isomorphic to S₃, Λ is a subgroup of index 6. The group Λ consists of all modular transformations for which a and d are odd and b and c are even.

Another important family of congruence subgroups are the modular group Γ₀(N) defined as the set of all modular transformations for which c ≡ 0 (mod N), or equivalently, as the subgroup whose matrices become upper triangular upon reduction modulo N. Note that Γ(N) is a subgroup of Γ₀(N). The modular curves associated with these groups are an aspect of monstrous moonshine – for a prime p, the modular curve of the normalizer is genus zero if and only if p divides the order of the monster group, or equivalently, if p is a supersingular prime; see details at congruence subgroup.