Arithmetic group

Definition
Suppose $$K$$ is a number field and $$\mathcal{O}$$ is the ring of integers in $$K$$. Suppose $$G$$ is a linear algebraic group over $$K$$. An arithmetic subgroup of $$G$$ is a subgroup of $$G$$ that is commensurable with the group $$G(\mathcal{O})$$. Here, $$G(\mathcal{O})$$ refers to the subgroup of $$G$$ comprising those matrices such that both the matrix and its inverse have all entries in $$\mathcal{O}$$.

An arithmetic group is a group realized as an arithmetic subgroup in a linear algebraic group.