Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-15T06:25:20.287Z Has data issue: false hasContentIssue false
  • English
  • Français

The Free Product of Two Groups with a Malnormal Amalgamated Subgroup

Published online by Cambridge University Press:  20 November 2018

A. Karrass  and
D. Solitar
A. Karrass
Affiliation:
York University, Toronto, Ontario
D. Solitar
Affiliation:
York University, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [1], B. Baumslag defined a subgroup U of a group G to be malnormal in G if gug–1 ∈ U, 1 ≠ u ∈ U, implies that g ∈ U. Baumslag considered the class of amalgamated products (A * B; U) in which U is malnormal in both A and B. These amalgamated products play an important role in the investigations of B. B. Newman [13] of groups with one defining relation having torsion. In this paper, we shall be concerned primarily with a generalization of this class.

Let U be a subgroup of a group G and let u ∈ U. Then the extended normalizer EG(u, U) of u relative to U in G is defined by

if u ≠ 1, and by EG(u, U) = U if u = 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 933 - 959
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

This research was supported by NRC Grants A5614 and A5602.

References

1. Baumslag, B., Generalized free products whose two-generator subgroups are free, J. London Math. Soc. 43 (1968), 601–606.Google Scholar
2. Baumslag, G., On generalized free products, Math. Z. 75 (1962), 423–438.Google Scholar
3. Greenberg, L., Discrete groups of motions, Can. J. Math. 12 (1960), 414–425.Google Scholar
4. Higman, G., Neumann, B. H., and Neumann, H., Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254.Google Scholar
5. Karrass, A., Magnus, W., and Solitar, D., Elements of finite order in groups with a single defining relation, Comm. Pure Appl. Math. 18 (1960), 57–66.Google Scholar
6. Karrass, A. and Solitar, D., The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227–255.Google Scholar
7. Karrass, A. and Solitar, D., Subgroups of HNN groups and groups with one defining relation, Can. J. Math. 28 (1971), 627–643.Google Scholar
8. Lewin, T., Finitely generated D-groups, J. Austral. Math. Soc. 7 (1967), 375–409.Google Scholar
9. Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory: Presentations of groups, Pure and Appl. Math., Vol. 13 (Interscience, New York, 1966).Google Scholar
10. Moldavanski, D. I., Certain subgroups of groups with one defining relation, Sibirsk. Mat. Z. 8 (1967), 1370–1384.Google Scholar
11. Neumann, B. H., An essay on free products with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–544.Google Scholar
12. Neumann, H., Generalized free products with amalgamated subgroups. II, Amer. J. Math. 71 (1949), 491–540.Google Scholar
13. Newman, B. B., Some aspects of one-relator groups, Ph.D. thesis, University of Canberra, 1969 (submitted to Acta Math.).Google Scholar
14. Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1964).Google Scholar