Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Question

Let's define $A$ and $B$ as follows:

$A$ = {e,a} $B$ = {e,b,2b}

Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to $\{\{e\},\{b\},\{2b\},\{a\},\{a+b\},\{a+2b\}\}$

This has 6 elements including the identity, but how do I know it's cyclic? I can see that if we have $a=3b$, then we get $\{\{e\},\{b\},\{2b\},\{3b\},\{4b\},\{5b\}\}$. Is that enough for the problem to be verified?

Answer 1

As Taylor said, just find a generator (and it's the obvious one. If still confused, I can give a hint).

In general, you can show that $\mathbb{Z}_m\times\mathbb{Z}_n\cong\mathbb{Z}_{mn}\iff \gcd(m,n)=1 $