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Lower bounds for the radical of the group algebra of a finite p-soluble group

Published online by Cambridge University Press:  20 January 2009

D. A. R. Wallace
D. A. R. Wallace
Affiliation:
University of Aberdeen.
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Over a field of characteristic p>0 the group algebra of a finite group has a unique maximal nilpotent ideal, the Jacobson radical of the algebra. The powers of the radical form a decreasing and ultimately vanishing series of ideals and it would be of interest to determine the least vanishing power. Apart from the work of Jennings (3) on p-groups little is known in general (cf. (5)) about this particular power of the radical (cf. Remarks of Brauer in (4), p. 144. Problem 15). In this paper we give non-trivial lower bounds for the index of the least vanishing power of the radical when the group is p-soluble. Of the lower bounds we give we show that that lower bound, which is dependent solely on the order of the group, is the best possible such bound and that this bound is invalid if the assumption of p-solubility is omitted.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 16 , Issue 2 , December 1968 , pp. 127 - 134
Copyright
Copyright © Edinburgh Mathematical Society 1968

References

REFERENCES

(1) Brauer, R. and Nesbitt, C., On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556–590.CrossRefGoogle Scholar
(2) Hall, M., The Theory of Groups (New York, 1959).Google Scholar
(3) 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
(4) Saaty, T. L. (edit.), Lectures on Modern Mathematics, vol. I (New York, 1963).Google Scholar
(5) Wallace, D. A. R., Group algebras with radicals of square zero, Proc. Glasgow Math Assoc. 5 (1962), 158–159.CrossRefGoogle Scholar