Hostname: page-component-7479d7b7d-k7p5g Total loading time: 0 Render date: 2024-07-13T12:46:59.721Z Has data issue: false hasContentIssue false
  • English
  • Français

The nilpotency index of the radicals of group algebras of finite groups whose Sylow 3-subgroups are extra-special of order 27 of exponent 3

Published online by Cambridge University Press:  14 November 2011

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

Synopsis

Let J(FG) be the Jacobson radical of the group algebra FG of a finite groupG with a Sylow 3-subgroup which is extra-special of order 27 of exponent 3 over a field F of characteristic 3, and let t(G) be the least positive integer t with J(FG)t = 0. In this paper, we prove that t(G) = 9 if G has a normal subgroup H such that (|G:H|, 3) = 1 and if H is either 3-solvable, SL(3,3) or the Tits simple group 2F4(2)'.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 1-2 , 1988 , pp. 117 - 132
Copyright
Copyright © Royal Society of Edinburgh 1988

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

1Carlson, J. F.. Module varieties and cohomology rings of finite groups. Vorlesungen aus dem Fachbereich Mathematik 13 (Essen: Essen University, 1985).Google Scholar
2Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.. Atlas of finite groups (Oxford: Clarendon Press, 1985).Google Scholar
3Dornhoff, L.. Group representation theory, parts A and B (New York: Dekker, 1971–1972).Google Scholar
4Feit, W.. The representation theory of finite groups (Amsterdam: North-Holland, 1982).Google Scholar
5Griess, R. L. Jr, and Lyons, R.. The automorphism group of the Tits simple group 2F4(2)'. Proc. Amer. Math. Soc. 52 (1975), 75–78.Google Scholar
6Hiss, G.. The modular characters of the Tits simple group and its automorphism group. Comm. Algebra 14 (1986), 125–154.Google Scholar
7Koshitani, S.. On the Loewy series of the group algebra of a finite p-solvable group with p-length> 1. Comm. Algebra 13 (1985), 2175–2198.CrossRefGoogle Scholar
8Koshitani, S.. On group algebras of finite groups. Representation theory II (Groups and orders). Lecture Notes in Mathematics 1178, pp. 109–128 (Berlin: Springer, 1986).Google Scholar
9Koshitani, S.. The Loewy structure of the projective indecomposable modules for SL(3,3) and its automorphism group in characteristic 3. Comm. Algebra 15 (1987), 1215–1253.Google Scholar
10Landrock, P.. Finite group algebras and their modules. London Math. Soc. LectureNote Series 84 (Edinburgh: Cambridge University Press, 1983).CrossRefGoogle Scholar
11Motose, K. and Ninomiya, Y.. On the nilpotency index of the radical of a group algebra. Hokkaido Math. J. 4 (1975), 261–264.CrossRefGoogle Scholar
12Wallace, D. A. R.. Group algebras with radicals of square zero. Proc. Glasgow Math. Assoc. 5 (1962), 158–159.CrossRefGoogle Scholar
13Wallace, 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
14Wilson, R. A.. The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits. Proc. London Math. Soc. (3) 48 (1984), 533–563.CrossRefGoogle Scholar