TITLE: Creating an additive structure over the set of all finite groups?
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: I believe that we can get a commutative, associative operation as follows. Let $G=A\times B$ and $H=A\times C$, where $B$ and $C$ have no common factor (when written as a direct product of indecomposable groups of order greater than one). Define $G+H=A\times D$, where $D$ is a product of cyclic groups of prime order such that $|D|=|B|+|C|$.