TITLE: Doubt about isomorphic groups!
QUESTION [5 upvotes]: Let $G$ and $H$ be two isomorphic groups.According to my present knowledge,both $G$ and $H$ have same properties and need not be distinguished.But my book says that $G$ and $H$ can behave differently if they are subgroups of a large group.I can't understand why it is ?

REPLY [2 votes]: Another example - let $G$ be any non-abelian group. Then the subgroups $H:=\{(g,1) : g \in G\}$ and $K=\{(g,g) : g \in G\}$ of the direct product $G \times G$ are isomorphic. However, $H$ is normal in $G \times G$, but $K$ is not.