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On five well-known commutator identities

Part of: Algebraic structures Foundations

Published online by Cambridge University Press:  09 April 2009

Graham J. Ellis
Graham J. Ellis
Affiliation:
University College Galway National University of Ireland Galway, Ireland
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Abstract

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We conjecture that five well-known identities universally satisfied by commutators in a group generate all such universal commutator identies. We use homological techniques to partially prove the conjecture.

MSC classification

Secondary: 20A05: Axiomatics and elementary properties 08A99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 1 , February 1993 , pp. 1 - 19
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Baer, R., ‘Representations of groups as quotient groups I-III’, Trans. Amer. Math. Soc. 58 (1945), 295–419.Google Scholar
[2]Brown, R., Johnson, D. L. and Robertson, E. F., ‘Some computations of non-abelian tensor products of groups,’ J. Algebra 3 (1987), 177–202.CrossRefGoogle Scholar
[3]Brown, R. and Loday, J.-L., ‘Van Kampen theorems for diagrams of spaces’, Topology 26 (1987), 311–335.CrossRefGoogle Scholar
[4]Brown, K. S., Cohomology of groups, Graduate Texts in Math. 87 (Springer-Verlag 1982).CrossRefGoogle Scholar
[5]Ellis, G. J., ‘Non-abelian exterior products of groups and exact sequences in the homology of groups,’ Glasgow Math. J. 29 (1987), 13–19.CrossRefGoogle Scholar
[6]Ellis, G. J., ‘The non-abelian tensor product of finite groups is finite’, J. Algebra 3 (1969), 469–482.Google Scholar
[7]Ellis, G. J. and Steiner, R. J., ‘Higher dimensional crossed modules and the homotopy groups of (n + 1)-ads’, J. Pure Appl. Algebra 46 (1987), 117–136.CrossRefGoogle Scholar
[8]Higgins, P. J. and ‘Baer invariants and the Birkoff-Witt theorem’, J. Algebra 11 (1969), 469–482.CrossRefGoogle Scholar
[9]Hilton, P. J. and Stammbach, U., A course in homological algebra, Graduate Texts in Math. 4 (Springer-Verlag 1971).CrossRefGoogle Scholar
[10]MacDonald, J., ‘Group derived functors’, J. Algebra 10 (1968), 448–477.CrossRefGoogle Scholar
[11]Miller, C., ‘The second homology group of a group’, Proc. Amer Math. Soc. 3 (1952), 588–595.CrossRefGoogle Scholar
[12]Magnus, W., Karass, A. and Solitar, D., Combinatorial group theory (Interscience Publishers 1966).Google Scholar
[13]Macdonald, I. D. and Neumann, B. H., ‘On commutator laws in groups 2’, in: Proc. Amer. Math. Soc. Special Session on Combinatorial Group Theory at Univ. Maryland, to appear.Google Scholar
[14]Neumann, H., Varieties of groups (Springer-Verlag Berlin 1967).CrossRefGoogle Scholar
[15]Serre, J.-P., Lie algebras and Lie groups, Lecture Notes, Harvard University (Benjamin, New York, 1965).Google Scholar
[16]Ward, M., ‘Bases for polynilpotent groups’, Proc. London Math. Soc. (3) 24 (1972), 409–431.CrossRefGoogle Scholar