How to add two numbers from a group theoretic perspective? It is known that adding two numbers and looking at the carrying operation has a link with cocycles in group theory. (https://www.jstor.org/stable/3072368?origin=crossref)
When we add two numbers by elementary addition, we choose a basis $b$ for example $b=2$ which corresponds to the cyclic group $C_2$.
Suppose we have words $w_1,w_2$ of (possibly different) lengths from this group $C_2$, how do we add them to get a new word $w$ in elementary addition?
For example: $2=10_2=w_1$, $3=11_2=w_2$. Consider these as $w_1$ and $w_2$. Adding these numbers we get $5=101_2$ so the new word $w=101$.
(1) But how exactly is the process of adding these two "words" from $C_2 = \{0,1\}$ in group theoretic means?
(2) Is this "elementary addition" also possible for example for a non-cylcic group such as the Klein Four group?
(3) We also assign a number to such a word (b-adic expansion). Is this assignment also possible for the Klein Four group?
Thanks for your help.
Edit: In view of the plot given below, I decided to put the tag "fractals" to this question.
 A: I think the point is that, forgetting the final carry, the group of $n$-digit binary words is isomorphic to $C_{2^n}$. In the simplest case, the group of 2-digit binary words is isomorphic to $C_4$, which is built as a nontrivial extension
$$ 0 \to C_2 \to C_4 \to C_2 \to 0 $$
The 2-cocycle you mention is the one corresponding to this extension. In general, $C_{2^n}$ is built up as an iterated extension of $C_2$'s in the same way, with each carry being the associated 2-cocycle.
If we want to avoid forgetting the final carry, we can take the limit of the whole system to get the 2-adics $\mathbb{Z}_2$. The natural numbers $\mathbb{N}$ sit inside this as the submonoid of "finite words" (words whose digits are eventually 0 as we read right to left)
A: Doc, the proper place for the Klein-4 group in elementary arithmetic is multiplication, not addition. Namely, it is the group if invertible modulo 8 integers. Thus, they will represent in binary as words $(a,b,1)$ and you can work out the multiplication table, but it ain't gonna be a big surprise...
