The group \(\operatorname{GL}_n(\mathbb{A}_{\mathbb{Q}})\).

The adeles \(\mathbb{A}_{\mathbb{Q}}\) of \(\mathbb{Q}\) are the product \(\mathbb{A}_{\mathbb{Q}}^f\times\mathbb{R}\), with the product topology. They are a topological ring. Hence \(\operatorname{GL}_n(\mathbb{A}_{\mathbb{Q}})=\operatorname{GL}_n(\mathbb{A}_{\mathbb{Q}}^f)\times\operatorname{GL}_n(\mathbb{R})\) is a topological group, where we are being a bit liberal with our use of the equality symbol.

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