Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-16T07:00:22.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Group algebras of finite p-solvable groups with radicals of the fourth power zero

Published online by Cambridge University Press:  14 November 2011

Shigeo Koshitani
Shigeo Koshitani
Affiliation:
Department of Mathematics, Faculty of Science, Chiba University, 1-33 Yayoi-cho, Chiba-city, 260, Japan
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let J(KG) be the Jacobson radical of the group algebra KG of a finite p-solvable group G over a field K of characteristic p > 0, and let t(G) be the least positive integer t such that J(KG)t = 0. In this paper we determine the structure of G with t(G) = 4 under the assumption that H is abelian, H is metacyclic, or the order of H is not divisible by 3 where H = O2'(G).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 3-4 , 1982 , pp. 205 - 231
Copyright
Copyright © Royal Society of Edinburgh 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Brauer, R.. Zur Darstellungstheorie der Gruppen endlicher Ordnung II. Math. Z. 72 (1959), 25–46.CrossRefGoogle Scholar
2Brauer, R.. Some applications of the theory of blocks of characters of finite groups III. J. Algebra 3 (1966), 225–255.CrossRefGoogle Scholar
3Brauer, R.. On blocks and sections in finite groups I.Amer. J. Math. 89 (1967), 1115–1136.CrossRefGoogle Scholar
4Brauer, R.. On the structure of blocks of characters of finite groups. Proceedings of the second international conference on the theory of groups 1973 (edited by Newman, M. F.). Lecture Notes in Mathematics 372, 103–130 (Berlin: Springer, 1974).Google Scholar
5Brauer, R.. On 2-blocks with dihedral defect groups. Symposia Mathematica 13 (1974), 367–393.Google Scholar
6Curtis, C. W. and Reiner, I.. Representation theory of finite groups and associative algebras (New York: Interscience, 1962).Google Scholar
7Dade, E. C.. Blocks with cyclic defect groups. Ann. of Math. 84 (1966), 20–48.CrossRefGoogle Scholar
8Dornhoff, L.. Group representation theory, parts A and B (New York: Dekker, 1971–1972).Google Scholar
9Fong, P.. On the characters of p–solvable groups. Trans. Amer. Math. Soc. 98 (1961), 263–284.Google Scholar
10Glauberman, G.. Correspondences of characters for relatively prime operator groups. Canad. J. Math. 20 (1968), 1465–1488.CrossRefGoogle Scholar
11Gorenstein, D.. Finite groups (New York: Harper and Row, 1968).Google Scholar
12Hall, P. and Higman, G.. On the p–length of p–soluble groups and reduction theorems for Burnside's problem. Proc. London Math. Soc. 6 (1956), 1–42.CrossRefGoogle Scholar
13Huppert, B.. Endliche Gruppen I (Berlin: Springer, 1967).CrossRefGoogle Scholar
14Isaacs, I. M.. Character theory of finite groups (New York: Academic Press, 1976).Google Scholar
15Jennings, 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
16Koshitani, S.. On the nilpotency indices of the radicals of group algebras of p–groups which have cyclic subgroups of index p. Tsukuba J. Math. 1 (1977), 137–148.CrossRefGoogle Scholar
17Michler, G. O.. Blocks and centers of group algebras. Lectures on rings and modules. Lecture Notes in Mathematics 246, 429–563 (Berlin: Springer, 1972).CrossRefGoogle Scholar
18Michler, G. O.. Green correspondence between blocks with cyclic defect groups I. J. Algebra 39 (1976), 26–51.CrossRefGoogle Scholar
19Morita, K.. On group rings over a modular field which possess radicals expressible as principal ideals. Sci. Rep. Tokyo Bunrika Daigaku A4 (1951), 177–194.Google Scholar
20Motose, K.. On the nilpotency index of the radical of a group algebra II. Math. J. Okayama Univ. 22 (1980), 141–143.Google Scholar
21Motose, K. and Ninomiya, Y.. On the nilpotency index of the radical of a group algebra. Hokkaido Math. J. 4 (1975), 261–264.CrossRefGoogle Scholar
22Passman, D. S.. Radicals of twisted group rings. Proc. London Math. Soc. 20 (1970), 409–437.CrossRefGoogle Scholar
23Schwarz, W.. Die Struktur modularer Gruppenringe endlicher Gruppen der p–Länge 1. J. Algebra 60 (1979), 51–75.CrossRefGoogle Scholar
24Wallace, D. A. R.. Group algebras with radicals of square zero. Proc. Glasgow Math. Assoc. 5 (1962), 158–159.CrossRefGoogle Scholar
25Wallace, D. A. R.. Lower bounds for the radical of the group algebra of a finite p–soluble group. Proc. Edinburgh Math. Soc. 16 (1968/1969), 127–134.CrossRefGoogle Scholar
26Yamazaki, K.. On projective representations and ring extensions of finite groups. J. Fac. Sci. Univ. Tokyo 10 (1964), 147–195.Google Scholar