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Group structure and properties of block ideals of the group algebra

Published online by Cambridge University Press:  18 May 2009

Wolfgang Hamernik
Wolfgang Hamernik
Affiliation:
Mathematisches Institut der Universität, D 63 Glessen, Arndtstr. 2, W. Germany
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In this note relations between the structure of a finite group G and ringtheoretical properties of the group algebra FG over a field F with characteristic p > 0 are investigated. Denoting by J(R) the Jacobson radical and by Z(R) the centre of the ring R, our aim is to prove the following theorem generalizing results of Wallace [10] and Spiegel [9]:

Theorem. Let G be a finite group and let F be an arbitrary field of characteristic p > 0. Denoting by BL the principal block ideal of the group algebra FG the following statements are equivalent:

(i) J(B1) ≤ Z(B1)

(ii) J(B1)is commutative,

(iii) G is p-nilpotent with abelian Sylowp-subgroups.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 16 , Issue 1 , March 1975 , pp. 22 - 28
Copyright
Copyright © Glasgow Mathematical Journal Trust 1975

References

REFERENCES

1.Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (New York, 1962).Google Scholar
2.Fong, P., On the characters of p-solvable groups, Trans. Amer. Math. Soc. 98 (1961), 263–284.Google Scholar
3.Hamernik, W., The linear character of an indecomposable module of a group algebra, J. London Math. Soc. (2) 7 (1973), 220–224.CrossRefGoogle Scholar
4.Koczian, S., Zerlegungseigenschaften von Gruppenringen, Thesis, University of Munich, (1973).Google Scholar
5.Michler, G. O., Blocks and centers of group algebras (in: Lectures on rings and modules (Springer lecture notes in mathematics 246, 430–552, Berlin etc. 1972).Google Scholar
6.Michler, G. O., The blocks of p-nilpotent groups over arbitrary fields, J. Algebra 24 (1973), 303–315.CrossRefGoogle Scholar
7.Morita, K., On group rings over a modular field which possess radicals expressible as principal ideals, Science reports of the Tokyo Bunrika Daigaku 4 (1951), 177–194.Google Scholar
8.Spiegel, H., Gruppenalgebren mit zentralem Radikal, Archiv d. Math. 23 (1972), 380–383.CrossRefGoogle Scholar
9.Spiegel, H., Gruppenalgebren mit Hauptblockidealen mit zentralem Radikal (to be published).Google Scholar
10.Wallace, D. A. R., On the commutativity of the radical of a group algebra, Proc. Glasgow Math. Assoc. 7 (1965), 1–8.CrossRefGoogle Scholar