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# On Certain Finitely Generated Subgroups of Groups Which Split

Published:2003-03-01
Printed: Mar 2003
• Myoungho Moon
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## Abstract

Define a group $G$ to be in the class $\mathcal{S}$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that $g^n \in K$ for all $g\in G$, $K$ has finite index in $G$. We show that a free product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of $C$ is finitely generated.
 Keywords: free product with amalgamation, $\HNN$ group, graph of groups, fundamental group
 MSC Classifications: 20E06 - Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E08 - Groups acting on trees [See also 20F65] 57M07 - Topological methods in group theory