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 2x2 integer matrices of determinant 1, such that the off-diagonal entries are *even*.

An important class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group SL(n, **Z**) we can reduce the entries to modular arithmetic in **Z**/N**Z** for any N >1, which gives a homomorphism

*SL*(*n*,**Z**) →*SL*(*n*,**Z**/*N*·**Z**)

of groups. The kernel of this reduction map is an example of a congruence subgroup – the condition is that the diagonal entries are congruent to 1 mod *N,* and the off-diagonal entries be congruent to 0 mod *N* (divisible by *N*), and is known as a **principal congruence subgroup**, Γ(*N*). Formally a congruence subgroup is one that contains Γ(*N*) for some *N*, and the least such *N* is the *level* or *Stufe* of the subgroup.

In the case *n=2* we are talking then about a subgroup of the modular group (up to the quotient by {I,-I} taking us to the corresponding projective group): the kernel of reduction is called Γ(N) and plays a big role in the theory of modular forms. Further, we may take the inverse image of any subgroup (not just {e}) and get a congruence subgroup: the subgroups Γ_{0}(N) important in modular form theory are defined in this way, from the subgroup of mod *N* *2x2* matrices with 1 on the diagonal and 0 below it.

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' respected by the subgroup, and so some general idea of what 'congruence' means.

Read more about Congruence Subgroup: Congruence Subgroups and Topological Groups, Congruence Subgroups of The Modular Group

### Famous quotes containing the word congruence:

“As for butterflies, I can hardly conceive

of one’s attending upon you; but to question

the *congruence* of the complement is vain, if it exists.”

—Marianne Moore (1887–1972)