Closed Subgroups of P-adic Integers
Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Then the closed subgroups of $\Z_p$ are the principal ideals:
- $\text a) \quad \set 0$
- $\text b) \quad \forall k \in \N : p^k \Z_p$
Proof
From Metric Space is Hausdorff:
- $\Z_p$ is a Hausdorff space
From Finite Subspace of Hausdorff Space is Closed:
- $\set 0$ is closed
From Cosets Form Local Basis of P-adic Number:
- $\forall k \in \N : p^k \Z_p = 0 + p^k \Z_p$ is closed
Hence the ideals:
- $\text a) \quad \set 0$
- $\text b) \quad \forall k \in \N : p^k \Z_p$
It remains to show that an arbitrary closed subgroups is indeed one of these ideals.
Let $H$ be a closed subgroup of the additive group of $\Z_p$.
From Correspondence between Abelian Groups and Z-Modules:
- $\Z H \subseteq H$
Let $a \in \Z_p$.
Let $h \in H$.
From Integers are Dense in P-adic Integers, there exists a sequence $\sequence {a_n}$:
- $\forall n \in \N: a_n \in Z$
- $\ds \lim_{n \mathop \to \infty} a_n = a$
From Multiple Rule for Sequences in Normed Division Ring:
- $\ds \lim_{n \mathop \to \infty} a_n h = a h$
We have:
- $a_n h \in \Z H \subseteq H$
From Subset of Metric Space contains Limits of Sequences iff Closed:
- $a h = \ds \lim_{n \mathop \to \infty} a_n h \in H$
Since $a \in \Z_p$ and $h \in H$ were arbitrary, it follows that:
- $\Z_p H \subseteq H$
By definition of ideal:
- $H$ is an ideal of $\Z_p$
From Ideals of P-adic Integers, $H$ is one of the principal ideals:
- $\text a) \quad \set 0$
- $\text b) \quad \forall k \in \N : p^k \Z_p$
$\blacksquare$