# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## The subgroups of a free product of two groups with an amalgamated subgroupHTML articles powered by AMS MathViewer

by A. Karrass and D. Solitar
Trans. Amer. Math. Soc. 150 (1970), 227-255 Request permission

## Abstract:

We prove that all subgroups \$H\$ of a free product \$G\$ of two groups \$A,B\$ with an amalgamated subgroup \$U\$ are obtained by two constructions from the intersection of \$H\$ and certain conjugates of \$A,B\$, and \$U\$. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-Neumann-Neumann group. The particular conjugates of \$A,B\$, and \$U\$ involved are given by double coset representatives in a compatible regular extended Schreier system for \$G\$ modulo \$H\$. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let \$A\$ and \$B\$ have the property \$P\$ and \$U\$ have the property \$Q\$. Then it is proved that \$G\$ has the property \$P\$ in the following cases: \$P\$ means every f.g. (finitely generated) subgroup is finitely presented, and \$Q\$ means every subgroup is f.g.; \$P\$ means the intersection of two f.g. subgroups is f.g., and \$Q\$ means finite; \$P\$ means locally indicable, and \$Q\$ means cyclic. It is also proved that if \$N\$ is a f.g. normal subgroup of \$G\$ not contained in \$U\$, then \$NU\$ has finite index in \$G\$.
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