Cartan invariants of group algebras of finite groups
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- by Shigeo Koshitani
- Proc. Amer. Math. Soc. 124 (1996), 2319-2323
- DOI: https://doi.org/10.1090/S0002-9939-96-03401-6
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Abstract:
We give a result on Cartan invariants of the group algebra $kG$ of a finite group $G$ over an algebraically closed field $k$, which implies that if the Loewy length (socle length) of the projective indecomposable $kG$-module corresponding to the trivial $kG$-module is four, then $k$ has characteristic 2. The proof is independent of the classification of finite simple groups.References
- Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
- Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
- P. Landrock, Finite group algebras and their modules, London Mathematical Society Lecture Note Series, vol. 84, Cambridge University Press, Cambridge, 1983. MR 737910, DOI 10.1017/CBO9781107325524
- Peter Landrock, The Cartan matrix of a group algebra modulo any power of its radical, Proc. Amer. Math. Soc. 88 (1983), no. 2, 205–206. MR 695241, DOI 10.1090/S0002-9939-1983-0695241-2
- G. O. Michler, Blocks and centers of group algebras, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. I), Lecture Notes in Math., Vol. 246, Springer, Berlin, 1972, pp. 429–563. MR 0332949
- Wolfgang Müller, Unzerlegbare Moduln über artinschen Ringen, Math. Z. 137 (1974), 197–226 (German). MR 352178, DOI 10.1007/BF01237391
- Tetsuro Okuyama, On blocks of finite groups with radical cube zero, Osaka J. Math. 23 (1986), no. 2, 461–465. MR 856899
- D. A. R. Wallace, Group algebras with radicals of square zero, Proc. Glasgow Math. Assoc. 5 (1962), 158–159 (1962). MR 140568, DOI 10.1017/S2040618500034523
- Peter J. Webb, The Auslander-Reiten quiver of a finite group, Math. Z. 179 (1982), no. 1, 97–121. MR 643050, DOI 10.1007/BF01173918
Bibliographic Information
- Shigeo Koshitani
- Affiliation: Department of Mathematics, Faculty of Science, Chiba University, Yayoi-cho, Chiba-city, 263, Japan
- MR Author ID: 202274
- Email: koshitan@math.s.chiba-u.ac.jp
- 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.
- Communicated by: Ronald M. Solomon
- © Copyright 1996 American Mathematical Society
- 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
Dedicated: Dedicated to Professor Takeshi Kondo on his 60th birthday