Congruence subgroup

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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 .


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 .


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