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Third Engel groups

Published online by Cambridge University Press:  17 April 2009

N. D. Gupta
Affiliation:
Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
M. F. Newman
Affiliation:
Mathematics Research Section Institute of Advanced Studies School of Mathematical SciencesAustralian National UniversityCanberra, ACT 2601Australia
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Abstract

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We present some new results on third Engel groups which are motivated by computer calculations but are not dependent on them. They include:

• for n > 2 every n-generator third Engel group is nilpotent of class at most 2n – 1;

• the fifth term of the lower central series of a third Engel group has exponent dividing 20;

• the subgroup generated by fifth powers of elements in a third Engel group is nearly centre-by-metabeliami;

and a normal form theorem for freest third Engel groups without elements of order 2.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bachmuth, S. and Mochizuki, H.Y., ‘Third Engel groups and the Macdonald-Neumann conjecture’, Bull. Austral. Math. Soc. 5 (1971), 379386.CrossRefGoogle Scholar
[2]Cohn, P.M., ‘A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group’, Proc. Cambridge Philos. Soc. 51 (1955), 401405.CrossRefGoogle Scholar
[3]Gupta, C.K., see p.104 of Narain Gupta [5]. (1976)CrossRefGoogle Scholar
[4]Gupta, Narain, ‘Third-Engel 2-groups are soluble’, Canad. Math. Bull. 15 (1972), 523524.CrossRefGoogle Scholar
[5]Gupta, Narain, Burnside groups and related topics, Lecture notes, University of Manitoba, 1976.Google Scholar
[6]Hall, Marshall Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
[7]Havas, George and Newman, M.F., ‘Applications of computers to questions like those of Burnside’, in Burnside Groups, Spritiger Lecture Notes in Mathematics 806, pp. 211230 (Springer-Verlag, Belin, Heidelberg, New York, 1980).CrossRefGoogle Scholar
[8]Heineken, Hermann, ‘Engelsche Elemente der Länge drei’, Illinois J. Math. 5 (1961), 681707.CrossRefGoogle Scholar
[9]Heineken, H., ‘Bounds for the nilpotency class of a group’, J. London. Math. Soc. 37 (1962), 456458.CrossRefGoogle Scholar
[10]Kappe, L.-C. and Kappe, W.P., ‘On three-Engel groups’, Bull. Austral. Math. Soc. 7 (1972), 391405.CrossRefGoogle Scholar
[11]Macdonald, I.D. and Neumann, B.H., ‘A third-Engel 5-group’, J. Austral. Math. Soc. VII (1967), 555569.CrossRefGoogle Scholar
[12]Newman, H., Varieties of Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37 (Springer-Verlag, Berlin, Heidelberg, New York, 1987).Google Scholar
[13]Newman, M.F., ‘Some varieties of groups’, J. Austral. Math. Soc. XVI (1973), 481494.CrossRefGoogle Scholar
[14]Tobin, S.J., ‘Groups with exponent four’, in Groups - St Andrew., 1981, LMS Lecture Note Series 71, pp. 81136, 1981.Google Scholar
[15]Vaughan-Lee, M.R., ‘Derived lengths of Burnside groups of exponent 4’, Quart. J. Math. Oxford (2) 30 (1979), 495504.CrossRefGoogle Scholar