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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- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 150 (1970), 227-255
- MSC: Primary 20.52
- DOI: https://doi.org/10.1090/S0002-9947-1970-0260879-9
- MathSciNet review: 0260879