Say $f:\mathbb{Z}/a\mathbb{Z} \to \mathbb{Z}/b\mathbb{Z}$ and $g:\mathbb{Z}/a\mathbb{Z}\to \mathbb{Z}/c\mathbb{Z}$ are homomorphisms. When can the pushout $W$ be cyclic?
I don't know where to begin to answer this. I can't think of anything besides that for $f,g$ to be well defined, it must be that $c$ and $b$ have to divide $a$. But then I still don't know if $W$ is going to be cyclic,or not.
Also, how would one go about finding $W$ explicitly, given some fixed integers, say, $a=12,b=6,c=4 \,$?