I was thinking about how to prove $\operatorname{Br}(K)\cong H^2(\operatorname{Gal}(\bar{K}/K),\bar{K}^\*)$ without having to introduce inductive limits and all the profinite stuff. So, I started wondering if the conditions of a direct system could be weakened for the category of abelian groups in a way that isomorphisms would be still preserved. This brought me to the following general question:


Let $G$ and $H$ be two abelian groups, not necessarily finite, $I$ an index set and $(G\_i) \_{i\in I}$ and $(H\_i)\_{i\in I}$ families of subgroups respectevely of $G$ and $H$ such that


(1) $\forall i\in I: G\_i \cong H\_i$ and


(2) $\bigcup\_{i\in I}G\_i=G$ and $\bigcup\_{i\in I}H\_i=H$.


**Question 1:** Can we conclude that $G\cong H$? 


**Question 2:** If yes, can we drop "abelian"?


**EDIT:** I forgot to mention that the $G\_i$ (and $H\_i$) are also assumed to be distinct subgroups.



When are unions of isomorphic groups isomorphic?
The answer is no. 


For a counterexample, let $G\_i=\mathbb{Z}$ be the integers and let $H\_i=\frac1i\mathbb{Z}$, for positive natural numbers $i$. The union $\bigcup\_i G\_i=\mathbb{Z}$, but $\bigcup\_i H\_i=\mathbb{Q}$. 


For the revised question, where you want $G\_i$ and $H\_i$ distinct, there are still counterexamples, such as $G\_i=i\mathbb{Z}$ and $H\_i=\frac1i\mathbb{Z}$. 




---


On a positive note, if you have a bit more coherence in your isomorphisms, then you can make the affirmative conclusion. That is, if we can find particular isomorphisms $\pi\_i:G\_i\cong H\_i$ which agree on their common domains, then they will build together into an isomorphism of $G$ and $H$. That is, what you want is not merely that $G\_i\cong H\_i$, but rather that the way that $G\_i$ sits inside $G$ is the same as the way $H\_i$ sits inside $H$. More generally, if $I$ is not just a naked index set, but is a directed set, such that when $i\lt j$ in this order then we have maps $G\_i\to G\_j$ and $H\_i\to G\_j$ and the isomorphisms $G\_i\cong H\_i$ make a commutative system, then the direct limit $G$ of the $G\_i$'s will be isomorphic to the direct limit $H$ of the $H\_i$'s by universal property arguments. 


