Define the 2-adic valuations of the integers $\Bbb{N}$ as a group.
If we take the set of possible 2-adic valuations of integers $\{2^{-n}:n\in\Bbb{N}_{\geq0}\}$
Then we can define a group on this using the function $g(n)$:
$g:\Bbb{N}\to\Bbb{Z}: g(2n)=n, g(2n-1)=-n$
And we will have the group $G=(2^{-n}:n\in(\Bbb{N},\circ))$
Where in $\Bbb{N}$, $a\circ b=g^{-1}(g(a)+g(b))$
So I'm trying to be clearer on what $\circ$ looks like in $G$ - in the most simple terms. Is the function easily defined in a nice algebraic form and what are examples of its application?
So if I want the group $G=(\{2^{-n}:n\in\Bbb{N}\},\ast)$ then I think I need to use the group isomorphism $h:(\Bbb{N},\circ)\to G: h(n)=2^{-n}$ and set:
$a\ast b=h(h^{-1}(a)\circ h^{-1}(b))$
Is there a nice algebraic form for this group function $\ast$?
A little background on the motivation, it turns out that if the Collatz conjecture is true, this group accounts for a large part of its structure and complexity as I allude to here.