Cartan invariants

of group algebras of finite groups

Author:
Shigeo Koshitani

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2319-2323

MSC (1991):
Primary 20C05, 20C20

DOI:
https://doi.org/10.1090/S0002-9939-96-03401-6

MathSciNet review:
1328356

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a result on Cartan invariants of the group algebra of a finite group over an algebraically closed field , which implies that if the Loewy length (socle length) of the projective indecomposable -module corresponding to the trivial -module is four, then has characteristic 2. The proof is independent of the classification of finite simple groups.

**[1]**W. Feit,*The representation theory of finite groups*, North-Holland, Amsterdam, 1982. MR**83g:20001****[2]**S. Koshitani,*Group algebras of finite -solvable groups with radicals of the fourth power zero*, Proc. Royal Soc. Edinburgh**92A**(1982), 205--231. MR**84b:2007****[3]**P. Landrock,*Finite group algebras and their modules*, London Math. Soc. Lecture Note Series Vol.**84**, Cambridge, 1983. MR**85h:20002****[4]**P. Landrock,*The Cartan matrix of a group algebra modulo any power of its radical*, Proc. Amer. Math. Soc.**88**(1983), 205--206. MR**84f:20010****[5]**G.O. Michler,*Blocks and centers of group algebras*, in Lectures on rings and modules, Springer Lecture Notes in Math. Vol.**246**, Springer, Berlin, 1972, pp.429--563. MR**48:11274****[6]**W. Müller,*Unzerlegbare Moduln über artinschen Ringen*, Math. Z.**137**(1974), 197--226. MR**50:4665****[7]**T. Okuyama,*On blocks of finite groups with radical cube zero*, Osaka J. Math.**23**(1986), 461--465. MR**87k:20024****[8]**D.A.R. Wallace,*Group algebras with radicals of square zero*, Proc. Glasgow Math. Assoc.**5**(1962), 158--159. MR**25:3986****[9]**P.J. Webb,*The Auslander-Reiten quiver of a finite group*, Math. Z.**179**(1982), 97--121. MR**83b:16029**

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Additional Information

**Shigeo Koshitani**

Affiliation:
Department of Mathematics, Faculty of Science, Chiba University, Yayoi-cho, Chiba-city, 263, Japan

Email:
koshitan@math.s.chiba-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-96-03401-6

Received by editor(s):
August 17, 1994

Received by editor(s) in revised form:
February 7, 1995

Additional Notes:
Supported in part by the Alexander von Humboldt Foundation, the Mathematical Prizes Fund, the University of Oxford and the Sasakawa Foundation.

Dedicated:
Dedicated to Professor Takeshi Kondo on his 60th birthday

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1996
American Mathematical Society