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Let $G=A\ast_C B$ be an amalgamated free product of two infinite groups such that $A,B$ do not have any finite index normal proper subgroups. Is it true that $G$ does not have any finite index normal proper subgroups?

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Yes.

Note that a group $G$ has a proper finite-index normal subgroup if and only if there exists a nontrivial homomorphism from $G$ to some finite group.

Consider any homomorphism $$\varphi\colon A * B \to F,$$ where $F$ is a finite group. By hypothesis, $\varphi(A)$ and $\varphi(B)$ must be trivial. Since $A$ and $B$ generate the domain, it follows that $\varphi$ is trivial. Thus $A* B$ cannot have any proper finite-index normal subgroups. Since $A*_C B$ is a quotient of $A*B$, the same must hold for $A*_C B$.

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