This seems like something that should be obviously true, yet when my friend asked me how I knew it, I couldn't come up with a proof. And now I'm wondering if it's possible that this could somehow turn out to be false.
Suppose I have groups $A$ and $A'$ with respective subgroups $B$ and $B'$, and bijections $\phi:A\to A'$ and $\varphi:B\to B'$.
Can I say that the sets $A/B$ and $A'/B'$ are in $1-1$ correspondence? I've found that what I thought would be the natural bijection $a + B\mapsto \phi(a) + B'$ need not even be well-defined. Unfortunately this kills a two page argument that took me a whole day to write. :( But all I need is a bijection, not an isomorphism. Can anyone help?