Editing Congruence subgroup (section)

=== Arithmetic groups ===
{{Main article | Arithmetic group}}

The notion of an arithmetic group is a vast generalisation based upon the fundamental example of {{tmath|1= \mathrm{SL}_d(\Z) }}. In general, to give a definition one needs a [[semisimple algebraic group]] <math>\mathbf G</math> defined over <math>\Q</math> and a faithful representation {{tmath|1= \rho }}, also defined over {{tmath|1= \Q }}, from <math>\mathbf G</math> into {{tmath|1= \mathrm{GL}_d }}; then an arithmetic group in <math>\mathbf G(\Q)</math> is any group <math>\Gamma \subset \mathbf G(\Q)</math> that is of finite index in the stabiliser of a finite-index sub-lattice in {{tmath|1= \Z^d }}.