Let $G$ be an infinite group, let $S\_0\subseteq G$ be a subgroup and suppose that ${\frak C}$ is a collection of subgroups of $G$ such that


1. $C \cong S\_0$ for all $C\in {\frak C}$, and
2. for all $C, C'\in {\frak C}$ we have $C\subseteq C'$ or $C'\subseteq C$.


It is easy to see that $\bigcup{\frak C}\subseteq G$ is a subgroup of $G$, but do we necessarily have $\bigcup{\frak C} \cong S\_0$?



Union of an ascending chain of subgroups in group $G$ isomorphic to subgroup $S_0\subseteq G$
This can fail on cardinality grounds. For instance if $G$ is a vector space of dimension $\aleph\_1$ and $\mathfrak{C}$ is a chain of $\aleph\_1$ subspaces each of dimension $\aleph\_0$, then all of the groups in $\mathfrak{C}$ are isomorphic but their union is not isomorphic to any of them.


