Glauberman-Watanabe corresponding $p$-blocks of finite groups with normal defect groups are Morita equivalent
HTML articles powered by AMS MathViewer
- by Morton E. Harris PDF
- Trans. Amer. Math. Soc. 357 (2005), 309-335 Request permission
Abstract:
Let $G$ be a finite group and let $A$ be a solvable finite group that acts on $G$ such that the orders of $G$ and $A$ are relatively prime. Let $b$ be a $p$-block of $G$ with normal defect group $D$ such that $A$ stabilizes $b$ and $D\leq C_{G}(A)$. Then there is a Morita equivalence between the block $b$ and its Watanabe correspondent block $W(b)$ of $C_{G}(A)$ given by a bimodule $M$ with vertex $\Delta D$ and trivial source that on the character level induces the Glauberman correspondence (and which is an isotypy by a theorem of Watanabe).References
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Richard Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung. II, Math. Z. 72 (1959/60), 25–46 (German). MR 108542, DOI 10.1007/BF01162934
- E. C. Dade, Isomorphisms of Clifford extensions, Ann. of Math. (2) 92 (1970), 375–433. MR 269750, DOI 10.2307/1970626
- Everett C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), no. 3, 241–262. MR 593823, DOI 10.1007/BF01161413
- Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
- George Glauberman, Correspondences of characters for relatively prime operator groups, Canadian J. Math. 20 (1968), 1465–1488. MR 232866, DOI 10.4153/CJM-1968-148-x
- Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
- M. E. Harris and M. Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite $p$-solvable groups, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3435–3453. MR 1911507, DOI 10.1090/S0002-9947-02-02990-2
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
- I. M. Isaacs and Gabriel Navarro, Character correspondences and irreducible induction and restriction, J. Algebra 140 (1991), no. 1, 131–140. MR 1114910, DOI 10.1016/0021-8693(91)90150-7
- Reinhard Knörr, Blocks, vertices and normal subgroups, Math. Z. 148 (1976), no. 1, 53–60. MR 401897, DOI 10.1007/BF01187868
- S. Koshitani, personal communication.
- Shigeo Koshitani and Gerhard O. Michler, Glauberman correspondence of $p$-blocks of finite groups, J. Algebra 243 (2001), no. 2, 504–517. MR 1850660, DOI 10.1006/jabr.2001.8777
- Burkhard Külshammer, Crossed products and blocks with normal defect groups, Comm. Algebra 13 (1985), no. 1, 147–168. MR 768092, DOI 10.1080/00927878508823154
- W. F. Reynolds, Blocks and normal subgroups of finite groups, Nagoya Math. J. 22 (1963), 15–32. MR 153729
- J.-P. Serre, “Corps locaux”, Hermann, Paris, 1962.
- Jacques Thévenaz, $G$-algebras and modular representation theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1365077
- Atumi Watanabe, The Glauberman character correspondence and perfect isometries for blocks of finite groups, J. Algebra 216 (1999), no. 2, 548–565. MR 1692989, DOI 10.1006/jabr.1998.7779
Additional Information
- Morton E. Harris
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: harris@math.umn.edu
- Received by editor(s): October 9, 2002
- Received by editor(s) in revised form: July 29, 2003
- Published electronically: April 27, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 309-335
- MSC (2000): Primary 20C20
- DOI: https://doi.org/10.1090/S0002-9947-04-03478-6
- MathSciNet review: 2098097