homomorphisms induce isomorphism Assume that we have the following commutative diagram with groups and homomorphisms 
where  $b$ and $c$ are injective homomorphisms
$\begin{array}[c]{ccc}
A&\stackrel{a}{\rightarrow}&B\\
\downarrow\scriptstyle{b}&&\downarrow\scriptstyle{c}\\
C&\stackrel{d}{\rightarrow}&D
\end{array}$
Is it true that $C/b(A) \cong D/c(B)$? Why?  

Hint: Let $A$ and $B$ both be trivial groups. What does your question ask in this case?
