# Congruence subgroup

From Encyclopedia of Mathematics

A subgroup of the general linear group over a ring with the following property: There exists a non-zero two-sided ideal of such that , where

that is, contains all matrices in that are congruent to the unit matrix modulo . More generally, a subgroup of a linear group of degree over is said to be a congruence subgroup if

for some non-zero two-sided ideal .

When

is said to be the principal congruence subgroup corresponding to . The concept of a congruence subgroup arose first for . It is particularly effective and important from the point of view of applications for a Dedekind ring in the case , where is an algebraic group defined over the field of fractions of .

#### References

[1] | H. Bass, J. Milnor, J.-P. Serre, "Solutions of the congruence subgroup problem for () and ()" Publ. Math. IHES , 33 (1967) pp. 421–499 |

**How to Cite This Entry:**

Congruence subgroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Congruence_subgroup&oldid=11756

This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article