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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.

  1. 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).

  2. Let $G$ and G' be two groups with the same order ($< \infty$). If their abelianizations are isomorphic, is it true that G\cong G'.

  3. 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).

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    @user5980: what do you mea$n$ by a ri$n$g homomorphism between two ideals? (Is this a homomorphism of not necessarily unital rings, i.e. a rng homomorphism?)2011-04-29

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The answer to #2 is no. For example, the dihedral group $D_4$ and the quaternion group $Q_8$ both have order $8$ and abelianization $(\mathbb{Z}/2\mathbb{Z})^2$. (This implies, among other things, that they have the same character table.) Group theory would be very boring if anything like this was true.