Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 5 down vote accepted


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 $A*_C B$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.