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Concept
Congruence subgroup
Summary
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.
The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite index are essentially congruence subgroups.
Congruence subgroups of 2×2 matrices are fundamental objects in the classical theory of modular forms; the modern theory of automorphic forms makes a similar use of congruence subgroups in more general arithmetic groups.
The simplest interesting setting in which congruence subgroups can be studied is that of the modular group .
If is an integer there is a homomorphism induced by the reduction modulo morphism . The principal congruence subgroup of level in is the kernel of , and it is usually denoted . Explicitly it is described as follows:
This definition immediately implies that is a normal subgroup of finite index in . The strong approximation theorem (in this case an easy consequence of the Chinese remainder theorem) implies that is surjective, so that the quotient is isomorphic to Computing the order of this finite group yields the following formula for the index:
where the product is taken over all prime numbers dividing .
If then the restriction of to any finite subgroup of is injective. This implies the following result:
If then the principal congruence subgroups are torsion-free.
The group contains and is not torsion-free.
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