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=21411
Congruence subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_subgroup&oldid=21411
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article