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An element $\alpha \in {\mathbb{Z}}_p$ topologically generates the additive group of p-adic integers exactly when $\alpha \not\equiv 0(\mathrm{mod}p)$, equivalently when it is a p-adic unit. Topological generation means that the closure of the cyclic subgroup $\mathbb{Z}\alpha$ is all of ${\mathbb{Z}}_p$.

For a p-adic unit, multiplication by $\alpha$ permutes every finite quotient group $\mathbb{Z}/p^n\mathbb{Z}$. Thus its integer multiples meet every coset $a+p^n{\mathbb{Z}}_p$. These cosets form a neighbourhood basis, so the multiples are dense.

If instead $p\mid \alpha$, every multiple lies in the proper closed subgroup $p{\mathbb{Z}}_p$, making density impossible. In particular, one topologically generates ${\mathbb{Z}}_p$, whereas $p$ does not. The generated algebraic subgroup can remain countable; it is its closure that fills the uncountable topological group.

A group is virtually cyclic when it has a cyclic subgroup of finite index. A finitely generated abelian group is virtually cyclic exactly when its free abelian rank is at most one. Finite groups qualify because their trivial cyclic subgroup has finite subgroup index. The rank-one case is the infinite finitely generated abelian group case.