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Permutational products of groups

Published online by Cambridge University Press:  09 April 2009

B. H. Neumann
B. H. Neumann
Affiliation:
The University, Manchester
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We deal with questions about the possible embeddings of two given groups A and B in a group P such that the intersection of A and B is a given subgroup H. The data, consisting of the “constituents” A and B with the “amalgamated” subgroup H, form an amalgam.1 According to a classical theorem of Otto Schreier [5], every amalgam of two groups can be embedded in a group F, the “free product of A and B with amalgamated subgroup H” or the “generalized free product” of the amalgam. This has the property that every group P in which the amalgam is embedded and which is generated by the amalgam, is a homomorphic image of it. Hence theorems on the existence of certain embedding groups P can be interpreted also as theorems on the existence of certain normal subgroups of F.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 3 , August 1960 , pp. 299 - 310
Copyright
Copyright © Australian Mathematical Society 1960

References

[1]Baer, R., Free sums of groups and their generalizations. An analysis of the associative law. Amer. J. Math. 71 (1949) 706–742.CrossRefGoogle Scholar
[2]Neumann, B. H., An essay on free products of groups with amalgamations. Phil. Trans. Roy. Soc. London (A) 246 (1954) 503–554.Google Scholar
[3]Neumann, B. H. and Neumann, Hanna, A remark on generalized free products. J. London Math. Soc. 25 (1950) 202–204.CrossRefGoogle Scholar
[4]Neumann, B. H. and Neumann, Hanna, A contribution to the embedding theory of group amalgams. Proc. London Math. Soc. (3) 3 (1953) 245–256.Google Scholar
[5]Schreier, Otto, Die Untergruppen der freien Gruppen. Abh. math. Sem. Univ. Hamburg 5 (1927) 161–183.CrossRefGoogle Scholar
[6]Wiegold, James, Nilpotent products of groups with amalgamations. Publ. Math. Debrecen 6 (1959) 131–168.CrossRefGoogle Scholar