I'm trying to form a ring (or ring-like structure) out of the set of all finite groups.

Has anyone created/encountered an operation "+" before with the follow properties.

Let $G_1, G_2$ be finite groups and |G| denote the size of G, then

$$ |G_1| + |G_2| = |G_1 + G_2|$$

Where the left hand side is addition amongst natural numbers and the right hand side is our "abstract group addition"

And let $\times$ denote the direct product. Then:

$$ G_1 \times ( G_2 + G_3) = (G_1 \times G_2) + (G_1 \times G_3) $$

The "+" ideally would be commutative and associative, but I'm not sure if such an operator can necessarily exist.

My goal is to try to denote a notion of "rational groups" and dedekind cuts to groups to see if I can create a "fractional group theory" so to speak.