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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
[2]Boone, W. W., Collins, D. J. and Matijasevic, Yu. V., ‘Embeddings into semigroups with only a few defining relations’, Proc. of the Second Scandinavian Logic Symposium 27–40. Studies in Logic and Foundations of Mathematics, (North-Holland Publ. Co., Amsterdam, 1971).Google Scholar
[3]Borisov, V. V., ‘Simple examples of groups with unsolvable world problem’, Math. Notes 6 (1969), 768–775, (Math. Zametki, 6 (1969), 521–532, Russian).CrossRefGoogle Scholar
[4]Britton, J. L., ‘The word problem’, Ann. of Math. (1963), 16–32.CrossRefGoogle Scholar
[5]Collins, D. J., ‘Word and conjugacy problems in groups with only a few defining relations’, Zeitschr. f. Math. Logik und Grundlagen d. Math. 15 (1969), 305–325.CrossRefGoogle Scholar
[6]Collins, D. J., ‘On a group embedding theorem of V. V. Borisov.’ To appear in Bull. Lond. Math. Soc.Google Scholar
[7]Higman, G., ‘Subgroups of finitely presented groups’, Proc. Roy. Soc. Series A 262 (1961), 455–475.Google Scholar
[8]Higman, G., Neumann, B. H. and Neumann, H., ‘Embedding theorems for groups’, J. London Math. Soc. 24 (1949), 247–256.CrossRefGoogle Scholar
[9]Magnus, W., ‘Das Identitätsproblem für Gruppen mit einer definierenden Relation,’ Math. Ann. 106 (1932), 295–307. (See also W. Magnus, A. Karass and D. Solitar, Combinatorial Group Theory, Interscience 1966, 252–278.)CrossRefGoogle Scholar
[10]Matijasevic, Yu. V., ‘Simple examples of undecidable associative calculi’, Soviet Math. 8 (1967), 555–557.Google Scholar