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The subgroups of a free product of two groups with an amalgamated subgroup


Authors: A. Karrass and D. Solitar
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
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0260879-9
Keywords: Amalgamated products, generalized free products, tree products, subgroup structure, indecomposable subgroups, presentations, Schreier system, Reidemeister-Schreier theory, compatible regular extended Schreier system, Higman-Neumann-Neumann groups, HNN groups, finitely generated intersection property, finitely presented subgroups, Kuroš subgroup theorem, double-ended cosets, locally indicable groups, simply decomposable groups
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