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An Existence Theorem for Generalized Direct Products with Amalgamated Subgroups

  • C. Y. Tang (a1)

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Generalized direct products with amalgamated subgroups were introduced by B. H. Neumann and Hanna Neumann in their joint paper (4). In general, we call a given collection of groups with specified subgroups amalgamated an amalgam of groups; if all groups are abelian we speak of an abelian amalgam. The group freely generated by the amalgam is called the abelian free sum of the amalgam provided it contains the amalgam isomorphically. The free abelian sum need not exist. Hence one of the problems is to find necessary and sufficient conditions for its existence.

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References

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1. Baer, R., Free sums of groups and their generalizations, Amer. J. Math., 71 (1949), 706742.10.2307/2372361
2. Neumann, B. H., An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London, Ser. A, 246 (1954), 503554.10.1098/rsta.1954.0007
3. Neumann, B. H. and Neumann, H., A remark on generalized free products, J. London Math. Soc, 25 (1950), 202204.10.1112/jlms/s1-25.3.202
4. Neumann, B. H. and Neumann, H., A contribution to the embedding theory of group amalgams, Proc. London Math. Soc. (3), 3 (1953), 245256.
5. Neumann, H., Generalized free sum of cyclical groups, Amer. J. Math., 72 (1950), 671685.10.2307/2372282
6. Neumann, H., On an amalgam of abelian groups, J. London Math. Soc, 26 (1951), 228232.10.1112/jlms/s1-26.3.228
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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