As an elementary example of isomorphic groups, I want to show my students the isomorphism between the additive group of integers $(\mathbb Z, +)$ and the multiplicative group $(S, \times)$, where $S = \{ 2^k \mid k \in \mathbb Z \}$. I am wondering if there is any kind of conventional notation for the set $S$. My first instinct is to denote it as $2^{\mathbb Z}$, in analogy with how we write $2\mathbb Z = \{ 2k \mid k\in Z\}$. But $2^{\mathbb Z}$ already has a conventional meaning, as the set of all functions $f: \mathbb Z \to 2$.
Is there a conventional name or notation for the multiplicative group I'm interested in?