Consider a group $G$ and a fiber product of the form $G\times_H G$. Is it true that such fiber product always have a subgroup isomorphic to $G$?
I'm tempted to say "Yes" because it's true if you consider the extremal cases $H=G$ and $H=\{e\}$. But I have no idea how to prove it in the general case. Thank you!
Edit by Derek Holt: Let me attempt to state the problem more explicitly.
Let $G$ and $H$ be groups and let $\phi_1,\phi_2:G \to H$ be surjective homomorphisms.
Is it always true that the subgroup $\{ (g_1,g_2) : \phi_1(g_1) = \phi_2(g_2) \}$ of $G \times G$ has a subgroup isomorphic to $G$?