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On the Magnus–Smelkin embedding

Published online by Cambridge University Press:  20 January 2009

J. Mccool∗
J. Mccool∗
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1
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The generalization of the Magnus embedding [7] proved by Smelkin [9] may bestated as follows. Let L be a free group freely generated by the set xi(i∈I), and let R be a normal subgroup of L with G = L/R. If V is any variety of groups and ∏ is the V-freegroup with free generating set the symbols [g, xi] (g∈G, i∈I), then L/V(R) is embeddedin the semidirect product ∏ ⋊ G (where the action of G on ∏ is given by h · [g, xi] = [hg, xi], for h, g ∈ G).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 1 , February 1987 , pp. 133 - 142
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Birman, J. S., Braids, links and mapping class groups, Ann. of Math. Stud. 82 (1974).Google Scholar
2.Comerford, L. P., Subgroups of small cancellation groups, J. London Math. Soc. 17 (1978), 422–424.CrossRefGoogle Scholar
3.Dunwoody, M. J., The Magnus embedding, J. London Math. Soc. 44 (1969), 115–117.CrossRefGoogle Scholar
4.Fox, R. H., Free differential calculus. III. Subgroups, Ann. of Math. 64 (1956), 407–419.CrossRefGoogle Scholar
5.Hilton, P. J. and Stammbach, U., A course in homological algebra (Graduate Texts in Math. 4, Springer 1971).CrossRefGoogle Scholar
6.The Kourovka Notebook: Unsolved problems in group theory (Seventh augmented edition) (AMS translations (2) volume 121).Google Scholar
7.Magnus, W., On a theorem of Marshall Hall, Ann. of Math. 40 (1939), 764–768.CrossRefGoogle Scholar
8.Remeslennikov, V. N. and SOKOLOV, V. G., Some properties of a Magnus embedding, Algebra i Logika 9 (1970), 566–578.Google Scholar
9.Smelkin, A. L., Wreath products and varieties of groups, Izv. Akad. Nauk. SSSR, Ser. Mat. 29 (1965), 149–170.Google Scholar
10.Smelkin, A. L., A remark on my paper “Wreath products and varieties of groups”, Izv. Akad. Nauk. SSSR Ser. Mat. 31 (1967), 443–444.Google Scholar
11.Zassenhaus, H. J., The theory of groups (New York, 1958).Google Scholar