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

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- by A. Karrass and D. Solitar PDF
- 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 $G$.## References

- Benjamin Baumslag,
*Intersections of finitely generated subgroups in free products*, J. London Math. Soc.**41**(1966), 673–679. MR**199247**, DOI 10.1112/jlms/s1-41.1.673 - Gilbert Baumslag,
*A remark on generalized free products*, Proc. Amer. Math. Soc.**13**(1962), 53–54. MR**133357**, DOI 10.1090/S0002-9939-1962-0133357-6 - Gilbert Baumslag,
*On generalised free products*, Math. Z.**78**(1962), 423–438. MR**140562**, DOI 10.1007/BF01195185 - Graham. Higman,
*The units of group-rings*, Proc. London Math. Soc. (2)**46**(1940), 231–248. MR**2137**, DOI 10.1112/plms/s2-46.1.231 - G. Higman,
*Subgroups of finitely presented groups*, Proc. Roy. Soc. London Ser. A**262**(1961), 455–475. MR**130286**, DOI 10.1098/rspa.1961.0132 - Graham Higman, B. H. Neumann, and Hanna Neumann,
*Embedding theorems for groups*, J. London Math. Soc.**24**(1949), 247–254. MR**32641**, DOI 10.1112/jlms/s1-24.4.247 - A. G. Howson,
*On the intersection of finitely generated free groups*, J. London Math. Soc.**29**(1954), 428–434. MR**65557**, DOI 10.1112/jlms/s1-29.4.428
A. Kuroš, - Wilhelm Magnus, Abraham Karrass, and Donald Solitar,
*Combinatorial group theory: Presentations of groups in terms of generators and relations*, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR**0207802** - D. I. Moldavanskiĭ,
*Certain subgroups of groups with one defining relation*, Sibirsk. Mat. Ž.**8**(1967), 1370–1384 (Russian). MR**0220810** - B. H. Neumann,
*An essay on free products of groups with amalgamations*, Philos. Trans. Roy. Soc. London Ser. A**246**(1954), 503–554. MR**62741**, DOI 10.1098/rsta.1954.0007 - Hanna Neumann,
*Generalized free products with amalgamated subgroups*, Amer. J. Math.**70**(1948), 590–625. MR**26997**, DOI 10.2307/2372201 - Hanna Neumann,
*Generalized free products with amalgamated subgroups. II*, Amer. J. Math.**71**(1949), 491–540. MR**30522**, DOI 10.2307/2372346
O. Schreier, - A. J. Weir,
*The Reidemeister-Schreier and Kuroš subgroup theorems*, Mathematika**3**(1956), 47–55. MR**80662**, DOI 10.1112/S0025579300000887

*The theory of groups*. Vol. 2, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1956. MR

**15**, 501; MR

**18**, 188.

*Die Untergruppen der freien Gruppen*, Abh. Math. Sem. Univ. Hamburg

**5**(1927), 161-183.

## Additional Information

- © 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