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The SQ-universality of some finitely presented groups

Published online by Cambridge University Press:  09 April 2009

Peter M. Neumann
Peter M. Neumann
Affiliation:
Queen' CollegeOxford OX1 4AWEngland
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Following a suggestion of G. Higman we say that the group G is SQ-universal if every countable group is embeddable in some factor group of G. It is a well-known theorem of G. Higman, B. H. Neumann and Hanna Neumann that the free group of rank 2 is sq-universal in this sense. Several different proofs are now available (see, for example, [1] or [9]). It is my intention to prove the LEmma. If H is a subgroup of finite index in a group G, then G is SQ-universal if and only if H is SQ-universal.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 1 , August 1973 , pp. 1 - 6
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Higman, G., Neumann, B. H. and Neumann, Hanna, ‘Embedding theorems for groups’, J. London Math. Soc. (1) 24 (1949), 247–254.CrossRefGoogle Scholar
[2]Hoare, A. H. M., Karrass, A. and Solitar, D., ‘Subgroups of infinite index in Fuchsian groups’, Math. Z. 125 (1972), 59–69.CrossRefGoogle Scholar
[3]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.CrossRefGoogle Scholar
[4]Kurosh, A.G., Theory of groups, Vol. II (Chelsea, New York, 1956).Google Scholar
[5]Levin, F., ‘Factor groups of the modular group’, J. London Math. Soc. (1) 43 (1968), 195–203.CrossRefGoogle Scholar
[6]McCool, James, ‘Embedding theorems for coutable groups’, Canad. J. Math. 22 (1970), 827–835.CrossRefGoogle Scholar
[7]Neumann, B. H., ‘Some remarks on infinite groups’, J. London Math. Soc. (1) 12 (1937), 120–127.CrossRefGoogle Scholar
[8]Neumann, B. H., ‘An essay on free products of groups with amalgamations’, Philos. Trans. Roy. Soc. Lon. (A) 246 (1954), 503–554.Google Scholar
[9]Neumann, B. H. and Neumann, Hanna, ‘Embedding theorems for groups’, J. London Math. Soc. (1) 34 (1959), 465–479.CrossRefGoogle Scholar
[10]Neumann, Hanna, ‘Generalized free products with amalgamated subgroups, II’, Amer. J. Math., 71 (1949), 491–540.CrossRefGoogle Scholar
[11]Schupp, Paul E., ‘Small cancellation theory over free products with amalgamation’, Math. Ann., 193 (1971), 255–264.CrossRefGoogle Scholar
[12]Zieschang, H., Vogt, E. and Coldewey, H.-D., Flächen und ebene diskontinuierliche Gruppen, Lecture Notes in Mathematics 122 (Springer, Berlin-Heidelberg-New York, 1970).Google Scholar