What do subgroups of $\mathbb{Z}_2(2^{\infty})$ look like?

Question

What do subgroups of the Prufer 2-group $\mathbb{Z}_2(2^{\infty})$ look like?

Let's think of $\mathbb{Z}_2(2^{\infty})$ as the dyadic rationals modulo $1$

EDIT: An example of a subgroup and its subgroup would be helpful.

While we're at it; is there a better way of writing the dyadic rationals modulo $1$?

Answer 1

I don't know whether it is better, or not, but you could write the dyadic rationals are the direct limit of infinite cyclic subgroups of the rational numbers,i.e., $$ \varinjlim \left\{2^{-i}\mathbb{Z}\mid i = 0, 1, 2, \dots \right\} $$ Concerning subgroups of Prüfer $p$-groups, have a look at this post:

Characterising subgroups of Prüfer $p$-groups.