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By definition, a discrete subgroup $\Gamma$ of a Lie group $G$ is a subgroup such that there is some open neighborhood $U$ of the identity $e$ such that $\Gamma \cap U = \{e\}$.

How to show that $\mathbb{Z}[1/p]$ is a discrete subgroup of $\mathbb{R} \times \mathbb{Q}_p$? Here $\mathbb{Z}[1/p]=\{\sum_{i=1}^n a_i/p^i \mid n \in \mathbb{Z}_{\geq 1}\}$ and $\mathbb{Q}_p$ is the field of $p$-adic numbers.

I think that first we need to embed $\mathbb{Z}[1/p]$ to $\mathbb{R} \times \mathbb{Q}_p$. Thank you very much.

LJR
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1 Answers1

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There's only one reasonable embedding (the diagonal embedding).

This is a obviously an injective homomorphism of topological groups, so it suffices to show that the origin has a neighborhood missing all the other points.

Consider a neighborhood of the form $(-\epsilon, \epsilon) \times p\mathbb{Z}_p$ where $\epsilon < 1$ is real. This is open and does the job (you need to check that nothing in $\mathbb{Z}[1/p]$ can both be close to 0 in the real topology and divisible by $p$.)

hunter
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