Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T20:22:39.100Z Has data issue: false hasContentIssue false

GROUP ALGEBRAS WHOSE GROUP OF UNITS IS POWERFUL

Published online by Cambridge University Press:  15 December 2009

VICTOR BOVDI*
Affiliation:
Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O.B. 12, Institute of Mathematics and Informatics, College of Nyíregyháza, Sóstói út 31/b, H-4410 Nyíregyháza, Hungary (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association, Inc. 2009

References

[1]Berger, T. R., Kovács, L. G. and Newman, M. F., ‘Groups of prime power order with cyclic Frattini subgroup’, Nederl. Akad. Wetensch. Indag. Math. 42(1) (1980), 1318.CrossRefGoogle Scholar
[2]Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D., Analytic Pro-p Groups, Cambridge Studies in Advanced Mathematics, 61 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[3]Hall, M. Jr, The Theory of Groups (Macmillan, New York, 1959).Google Scholar
[4]Héthelyi, L. and Lévai, L., ‘On elements of order p in powerful p-groups’, J. Algebra 270(1) (2003), 16.CrossRefGoogle Scholar
[5]Lubotzky, A. and Mann, A., ‘Powerful p-groups. I. Finite groups’, J. Algebra 105(2) (1987), 484505.CrossRefGoogle Scholar
[6]Newman, M. F., ‘On a class of nilpotent groups’, Proc. London Math. Soc. (3) 10 (1960), 365375.CrossRefGoogle Scholar
[7]Wilson, L., ‘On the power structure of powerful p-groups’, J. Group Theory 5(2) (2002), 129144.CrossRefGoogle Scholar