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Let $G$ be a group with two homomorphism $\phi \colon G \rightarrow H$ and $\psi \colon G \rightarrow F$.

If $\phi$ is an isomorphism, is it true that $F \simeq H \ast_G F$, the free product with amalgamation?

Shaun
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user73577
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    You've been here long enough to know that this is not how one asks questions on MSE. Please provide context. – Shaun Jul 11 '18 at 21:10

1 Answers1

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Hint: We have a commutative diagram $$ \require{AMScd}\begin{CD} G @> \phi >> H \\ @V \psi VV @VV \psi \circ \phi^{-1} V \\ F @> \operatorname{id} >> F. \end{CD} $$ Can you show that this diagram satisfies the universal property for the amalgamated product?

Shaun
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Daniel Schepler
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  • This does not work for infinite groups. – Shaun Jul 11 '18 at 21:51
  • @Shaun Why not? (I guess I am making the implicit assumption that the amalgamated product desired is using the given morphisms $\phi$ and $\psi$, and not possibly using some other random maps between the groups unrelated to $\phi$ or $\psi$ as your answer did. My interpretation seems to me like the most likely intended meaning, though it is slightly ambiguous as stated.) – Daniel Schepler Jul 11 '18 at 21:58
  • It does work, the universal property for pushout diagrams is independent of the carnidal of the objects in the category. – user73577 Jul 11 '18 at 21:58
  • Because my answer is a counterexample. – Shaun Jul 11 '18 at 21:59
  • The fundamental group of the Klein bottle is not $\Bbb Z$. – Shaun Jul 11 '18 at 22:00
  • Is not, your $\phi$ is not surjective. – user73577 Jul 11 '18 at 22:00
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    Shaun's answer is a counterexample to: if $G \simeq H$ but $\phi : G \to H$ is not itself necessarily an isomorphism, then the pushout of $\phi$ and $\psi : G \to F$ is not necessarily isomorphic to $F$. My answer is hinting at the problem: if $\phi$ is an isomorphism then the pushout of $\phi$ and $\psi$ is isomorphic to $F$. Two different questions with a slightly subtle difference between them. – Daniel Schepler Jul 11 '18 at 22:03
  • Oh, of course! Now I see it. I'm sorry. – Shaun Jul 11 '18 at 22:05