Hostname: page-component-7479d7b7d-pfhbr Total loading time: 0 Render date: 2024-07-12T05:38:52.642Z Has data issue: false hasContentIssue false
  • English
  • Français

On the embedding of a group in a join of given groups

Published online by Cambridge University Press:  09 April 2009

P. Hall
P. Hall
Affiliation:
50, Impington Lane, Histon, Cambs, England
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.

1. In their fundamental paper of 1949, Higman, Neumann and Neumann proved for the first time that a countable group can always be embedded in some 2-generator group: [1], Theorem IV. Two kinds of improvement of this result have recently appeared. In [4], Theorem 2, Dark has shown that the embedding can always be made subnormally. On the other hand, in [2], Theorem 2.1, Levin has shown that the two generators can be given preassigned orders m > 1 and n > 2; and in [3], Miller and Schupp prove that the 2-generator group can also be made to satisfy several additional requirements, such as being complete and Hopfian.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 17 , Issue 4 , June 1974 , pp. 434 - 495
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Higman, Graham, Neumann, B. H. and Neumann, Hanna, ‘Embedding theorems for groups’, J. London. Math. Soc. 24 (1949), 247–254.CrossRefGoogle Scholar
[2]Levin, F., ‘Factor groups of the modular groups’, J. London. Math. Soc. 43 (1968), 195–203.CrossRefGoogle Scholar
[3]Miller, Charles F. III and Schupp, Paul E., ‘Embeddings into Hopfian groups’, J. of Algebra 17 (1971), 171–176.CrossRefGoogle Scholar
[4]Dark, R. S., ‘On subnormal embedding theorems for groups’, J. London. Math. Soc. 43 (1968), 387–390.CrossRefGoogle Scholar
[5]Neumann, B. H., ‘An Essay on free products of groups with amalgamations’, Phil. Trans. Royal Soc. London Ser. A, 246 (1954), 503–554.Google Scholar
[6]Onofri, L., ‘Teoria delle sostituzioni che operano su una infinità numerabile di elemente, III’, Annali di Mat. (4) 7 (1929), 103–130.CrossRefGoogle Scholar
[7]Schreier, J. and Ulam, S., ‘Über die Permutationsgruppe der natürlichen Zahlenfolge’, Studia Math. 4 (1933), 134–141.CrossRefGoogle Scholar
[8]Baer, R., ‘Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich’, Studia Math. 5 (1935), 15–17.CrossRefGoogle Scholar
[9]Neumann, Hanna, Varieties of Groups, (Springer-Verlag, 1967).CrossRefGoogle Scholar
[10]Higman, Graham, ‘A finitely generated infinite simple group’, J. London Math. Soc. 26 (1951), 61–64.CrossRefGoogle Scholar
[11]Neumann, B. H. and Neumann, Hanna, ‘Embedding theorems for groups’, J. London Math. Soc. 34 (1959) 465–479.CrossRefGoogle Scholar
[12]Neumann, Hanna, ‘Generalized free products with amalgamated subgroups’, Amer. J. of Math. 70 (1948), 590–625.CrossRefGoogle Scholar
[13]Schupp, P. E., ‘A Survey of Small Cancellation Theory’, unpublished.Google Scholar
[14]Hall, Marshall Jr.The Theory of Groups (Macmillan Co. of New York, 1959).Google Scholar
[15]Huppert, B., Endliche Gruppen I, (Springer-Verlag, 1967).CrossRefGoogle Scholar
[16]Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups, (Springer-Verlag, 1957).CrossRefGoogle Scholar
[17]Camm, Ruth, ‘Simple free products’, J. London Math. Soc. 28 (1953), 66–76.CrossRefGoogle Scholar
[18]Wielandt, Helmut, Finite Permutation Groups, (Academic Press, 1964).Google Scholar
[19]Burnside, W., Theory of Groups of Finite Order, (Cambridge2nd ed. 1911).Google Scholar
[20]Hall, P., ‘On the finiteness of certain soluble groups’, Proc. London Math. Soc. (3) 9 (1959), 595–622.CrossRefGoogle Scholar
[21]Wielandt, H. and Huppert, B., ‘Arithmetical and normal structure of finite groups’, Proc. Symposia Pure Math. VI (1962), Amer. Math. Soc.CrossRefGoogle Scholar
[22]Schenkman, E., Group Theory, (Van Nostrand, 1965).Google Scholar
[23]Kaloujnine, Marc Kranser et Léo, ‘Produit complet des groupes de permutations et problème d'extension de groupes III’, Acta.Sci. Math. Szeged 14 (1951), 69–82.Google Scholar
[24]Kaloujnine, Léo, ‘La structure des p-groupes de Sylow des groupes symétriques finis’, Ann. Sci. Ecole Norm. Sup. (3) 65 (1948), 239–276.CrossRefGoogle Scholar
[25]Neumann, B. H. and Yamanuro, Sadayuki, ‘Boolean Powers of simple groups’, J. Austral. Math. Soc. 5. (1965), 315–324.CrossRefGoogle Scholar