I want to show that $\Bbb Z_2 +\Bbb Z_3$ is isomorphic to $\Bbb Z_6$ where '$+$' stands for external direct product.
First of all I wrote all the six elements of $\Bbb Z_2 + \Bbb Z_3$ and found the orders of each element. I observe that $\Bbb Z_2 +\Bbb Z_3$ and $\Bbb Z_6$ have same number of elements with number of elements of same order equal. What mapping should I define here in a valid way to show isomorphism?
I already know a theorem that when $\gcd(m,n)=1$, $\Bbb Z_m +\Bbb Z_n$ is isomorphic to $\Bbb Z_{mn}$. Shall I mention this theorem and let it go? Or other way is possible to show this?