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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 cathegory of abelian groups in a way that isomorphisms would be still preserved. This brought me to the following very 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.
show/hide this revision's text 2 added 103 characters in body

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. This brought me to the following very 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.
show/hide this revision's text 1

When are unions of isomorphic groups isomorphic?

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. This brought me to the following very 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"?