The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup

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Abstract
We prove that all subgroups of a free product of two groups with an amalgamated subgroup are obtained by two constructions from the intersection of and certain conjugates of , and . 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 , and involved are given by double coset representatives in a compatible regular extended Schreier system for modulo . The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let and have the property and have the property . Then it is proved that has the property in the following cases: means every f.g. (finitely generated) subgroup is finitely presented, and means every subgroup is f.g.; means the intersection of two f.g. subgroups is f.g., and means finite; means locally indicable, and means cyclic. It is also proved that if is a f.g. normal subgroup of not contained in , then has finite index in .

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