Congruence subgroup

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a congruence subgroup of a <a href="/facts/Matrix_group/wVzG0sa3">matrix group</a> with <a href="/facts/Integer/PUwwrawx">integer</a> entries is a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> defined by congruence conditions on the entries. A very simple example is the subgroup of <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> 2 × 2 integer matrices of <a href="/facts/Determinant/eGYqJ2TF">determinant</a> 1 in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for <a href="/facts/Arithmetic_subgroup/I8eA2483">arithmetic subgroups</a> of <a href="/facts/Algebraic_group/DDtBCIvh">algebraic groups</a>; 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 <a href="/facts/Residually_finite/SlY6WD2h">residually finite</a>. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a> are essentially congruence subgroups.
Congruence subgroups of 2 × 2 matrices are fundamental objects in the classical theory of <a href="/facts/Modular_form/PcnbYR1f">modular forms</a>; the modern theory of <a href="/facts/Automorphic_form/H0Pn7MyP">automorphic forms</a> makes a similar use of congruence subgroups in more general arithmetic groups.

Congruence subgroup open-in-new

Congruence subgroup