Arithmetic subgroup

Arithmetic groups

If ${\displaystyle \mathrm {G} }$ is an algebraic subgroup of ${\displaystyle \mathrm {GL} _{n}(\mathbb {Q} )}$ for some ${\displaystyle n}$ then we can define an arithmetic subgroup of ${\displaystyle \mathrm {G} (\mathbb {Q} )}$ as the group of integer points ${\displaystyle \Gamma =\mathrm {GL} _{n}(\mathbb {Z} )\cap \mathrm {G} (\mathbb {Q} ).}$ In general it is not so obvious how to make precise sense of the notion of "integer points" of a ${\displaystyle \mathbb {Q} }$-group, and the subgroup defined above can change when we take different embeddings ${\displaystyle \mathrm {G} \to \mathrm {GL} _{n}(\mathbb {Q} ).}$