Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-20T12:32:07.096Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups in which every Finitely Generated Subgroup is almost a Free Factor

Published online by Cambridge University Press:  20 November 2018

A. M. Brunner  and
R. G. Burns
A. M. Brunner
Affiliation:
University of Wisconsin—Park side, Kenosha, Wisconsin
R. G. Burns
Affiliation:
York University, Downsview, 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 [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 6 , 01 December 1979 , pp. 1329 - 1338
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Burns, R. G., A note on free groups, Proc. Amer. Math. Soc. 23 (1969), 14–17.Google Scholar
2. Burns, R. G., On finitely generated subgroups of free products, J. Austral. Math. Soc. 12 (1971), 358–364.Google Scholar
3. Cohen, D. E., Groups of cohomological dimension one, Lecture Notes in Mathematics, Vol. 245 (Springer-Verlag, 1972).Google Scholar
4. Dunwoody, M. J., Accessibility and groups of cohomological dimension one, (preprint).Google Scholar
5. Hall, M., Jr., Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421–432.Google Scholar
6. Huppert, B., Endliche gruppen I (Springer-Verlag, 1967).Google Scholar
7. Karrass, A. and Solitar, D., On finitely generated subgroups of a free group, Proc. Amer. Math. Soc. 22 (1969), 209–213.Google Scholar
8. 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
9. Karrass, A. and Solitar, D., The free product of two groups with a malnormal amalgamated subgroup, Can. J. Math. 23 (1971), 933–959.Google Scholar
10. Karrass, A., Pietrowski, A. and Solitar, D., Finite and infinite cyclic extensions of free groups, J. Austral. Math. Soc. 16 (1973), 458–466.Google Scholar
11. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory: Presentations of groups in terms of generators and relations, Pure and Appl. Math., vol. 13 (Interscience, 1966).Google Scholar
12. Romanovskiĭ, N. S., On the finite residuality relative to an embedding, of free products, Izv. Akad. Nauk SSSR, Ser. Matem. 33 (1969), 1324–1329.Google Scholar
13. Tretkoff, Marvin, Covering space proofs in combinatorial group theory, Communications in Algebra. 3 (1975), 429–457.Google Scholar