Here I want to ask some true or false problems regarding isomorphisms (and if false, is there some extra condition to make it true). I do not what is the correct title for these problem. And I also want some brief proofs if possible.
Let $R$ and R' be two rings with |R| =|R'|< \infty. And each proper ideal in $R$ is isomorphic to some ideal in R'. Then is it true that R\cong R'? (and if false, is there some extra condition to make it true).
Let $G$ and G' be two groups with the same order ($< \infty$). If their abelianizations are isomorphic, is it true that G\cong G'.
Let $\mathcal{D}$ be a subcategory of a category $\mathcal{C}$. (1) If $u$ is an isomorphism of $\mathcal{D}$, is $u$ an isomorphism of $\mathcal{C}$? (2) If $v$ is an isomorphism of $\mathcal{C}$, is $v$ an isomorphism of $\mathcal{D}$? (and if false, is there some extra condition to make it true).