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The class of a nilpotent wreath product

Published online by Cambridge University Press:  17 April 2009

David Shield
David Shield*
Affiliation:
Goroka Teachers College, Goroka, Papua New Guinea.
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Abstract

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Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.

A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.

Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 17 , Issue 1 , August 1977 , pp. 53 - 89
Copyright
Copyright © Australian Mathematical Society 1977

References

[1] Baumslag, Gilbert, “Wreath products and p-groups”, Proc. Cambridge Philos. Soc. 55 (1959), 224–231.CrossRefGoogle Scholar
[2] Bayes, A.J., Kautsky, J. and Wamsley, J.W., “Computation in nilpotent groups (application)”, Proc. Second Internat. Conf. Theory of Groups, Canberra 1973, 82–89 (Lecture Notes in Mathematics, 372. Springer-Verlag, Berlin, Heidelberg, New York, 1974).Google Scholar
[3] Buckley, Joseph T., “Polynomial functions and wreath products”, Illinois J. Math. 14 (1970), 274–282.CrossRefGoogle Scholar
[4] Hall, P., “A contribution to the theory of groups of prime-power order”, Proc. London Math. Soc. (2) 36 (1934), 29–95.CrossRefGoogle Scholar
[5] Havas, George, Wall, G.E., and Wamsley, J.W., “The two generator restricted Burnside group of exponent five”, Bull. Austral. Math. Soc. 10 (1974), 459–470.CrossRefGoogle Scholar
[6] Huppert, B., Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[7’ Jennings, S.A., “The structure of the group ring of a p-group over a modular field”, Trans. Amer. Math. Soc 50 (1941), 175–185.Google Scholar
[8’ HOCTPиHиH, A.и. [Kostrikin, A.I.], “O npoблeme БepHcaйдa” [The Burnside problem], Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 3–34; Amer. Math. Soc. Transl. (2) 36 (1964), 63–99Google Scholar
[9] Liebeck, Hans, “Concerning nilpotent wreath products”, Proc. Cambridge Philos. Soc. 58 (1962), 443–451.CrossRefGoogle Scholar
[10] Meldrum, J.D.P., “On nilpotent wreath products”, Proc. Cambridge Philos. Soc. 68 (1970), 1–15.CrossRefGoogle Scholar
[11] Meldrum, J.D.P., “Correction to a paper on wreath products”, Proa. Cambridge Philos. Soc. 76 (1974), 21.CrossRefGoogle Scholar
[12] Morley, Larry, “Bounds on the nilpotency class of certain semidirect products”, Trans. Amer. Math. Soc. 159 (1971), 381–390.CrossRefGoogle Scholar
[13] Morley, Larry J., “A commutator relation in a wreath product of p-groups”, preliminary report, Notices Amer. Math. Soc. 22 (1975), A–675, abstract no. 728–A31.Google Scholar
[14] Neumann, Hanna, Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 37. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[15] Sandling, Robert, “Modular augmentation ideals”, Proc. Cambridge Philos. Soc. 71 (1972), 25–32.CrossRefGoogle Scholar
[16] Scruton, Teresa, “Bounds for the class of nilpotent wreath products”, Proc. Cambridge Philos. Soc. 62 (1966), 165–169.CrossRefGoogle Scholar
[17] Shield, David, “Power and commutator structure of groups”, Bull. Austral. Math. Soc. 17 (1977), 1–52.CrossRefGoogle Scholar
[18] Sylow, L., “Théorèmes sur les groupes de substitutions”, Math. Ann. 5 (1872), 584–594.CrossRefGoogle Scholar