What is the difference -- product/direct sum of cyclic groups 
Possible Duplicate:
The direct sum $\oplus$ versus the cartesian product $\times$ 

Is there a difference in the proof in showing that two cyclic groups $C_m\times C_n\cong C_{mn}$ where $m,n$ are relatively prime and showing that $C_m\bigoplus C_n$ for the same $m,n$? Is there a catch? 
 A: So far as I know, the two notations mean the same thing. There is a difference if, instead of two groups, you have infinitely many. 
A: In any category, a coproduct $A\oplus B$ is characterized by having two maps $A\to A\oplus B$ and $B\to A\oplus B$ with the property that given any pair of maps $f: A\to C$ and $g: B\to C$ there exists a unique map $f\oplus g: A\oplus B \to C$ such that $A\to A\oplus B \to C$ is the map $f: A\to C$ and $B\to A\oplus B \to C$ is the map $g: B\to C$. 
Dually, a product $A\times B$ is characterized by having two maps $A\times B \to A$ and $A\times B \to B$ with the property that given any pair of maps $f: C\to A$ and $g: C\to B$ there exists a unique map $f\times g: C\to A\times B$ such that $C\to A\times B \to A$ is $f$ and $C\to A\times B \to B$ is $g$.
In an additive category (in which zero maps make sense), the universal properties give a canonical map $A\oplus B \to A\times B$ given by $\Gamma = (id_A\times 0) \oplus (0\times id_B)$. We also have the composite maps $\alpha: A\times B \to A \to A\oplus B$ and $\beta: A\times B \to B \to A\oplus B$, and since we can add maps we get a map $\Delta = \alpha + \beta: A\times B \to A\oplus B$. Now, we can use the universal properties to see that $\Gamma$ and $\Delta$ are inverse isomorphisms. 
