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Embeddings into groups with only a few defining relations

Published online by Cambridge University Press:  09 April 2009

W. W. Boone  and
D. J. Collins
W. W. Boone
Affiliation:
University of Illinois at Urbana-Champaign, U.S.A.
D. J. Collins
Affiliation:
Queen Mary College, University of London, England
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It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 1 , August 1974 , pp. 1 - 7
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Aanderaa, S., ‘A proof of Higman's embedding theorem using Britton extensions of groups’, Word Problems: decision problems and the Burnside problem in group theory 1–18. Studies in Logic and Foundations of Mathematics, (North-Holland Publ. Co., Amsterdam, 1973).Google Scholar
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