Does this isomorphism hold? Proposition: If $B\cong C$ then  $\dfrac{A\oplus B}{C} \cong A.$
This is clearly true for vector spaces by counting the dimensions, but I am most interested to see if it holds for groups. What about R-modules (R commutative ring)?
 A: Take the special case where $A$ is trivial, and your theorem becomes: if $B \cong C$ then $B/C$ is trivial.
This special case is true for finite groups (by counting) but not for infinite groups. Many infinite groups have proper subgroups to which they are isomorphic.
When you say $\displaystyle\frac{A \oplus B}{C}$ you are viewing $C$ as a subgroup of $B$, and in particular what you get depends on how you make that identification – you need to choose a particular embedding of $C$ into $B$. When you say $B \cong C$ you are making no such choice, and in particular when a group has lots of subgroups that are isomorphic, the LHS doesn't make any distinction but the RHS does. So you'd sort of expect it to be false (but, of course, vector spaces have so much structure that the problems go away).
A: No, it doesn't hold in general. For example $\mathbb{Z}\cong (0,2\mathbb{Z})$ as (abelian) groups (and hence also as $\mathbb{Z}$-modules). But e.g. $(\mathbb{Z}\oplus \mathbb{Z})/(0,2\mathbb{Z})\cong \mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}\ncong \mathbb{Z}$.
A: Ben Millwood and Julian Kuelshammer give counter-examples in the realm of infinite groups, by playing on the fact that infinite groups can have isomorphic, normal subgroups (i.e. there exists a group $G$ with $H$ a proper, normal subgroup and $G\cong H$). However, the result also fails for finite groups.
A finite groups counter example would be the direct product of the cyclic group of order $4$ with the cyclic group of order $2$, $G=C_{4} \times C_2$. As $C_4$ contains a cyclic subgroups of order $2$ we can quotient this out to get $C_2\times C_2\not\cong C_4$, as required.
A related, and rather interesting, fact is that there exists a (non-trivial) group $G$ such that $G\cong G\times G$. The proof can be found in the paper "Some odd finitely presented groups" by G. Baumslag and Miller III.
