A block theoretic analogue of a theorem of Glauberman and Thompson
Authors:
Radha Kessar and Markus Linckelmann
Journal:
Proc. Amer. Math. Soc. 131 (2003), 35-40
MSC (2000):
Primary 20C20
DOI:
https://doi.org/10.1090/S0002-9939-02-06506-1
Published electronically:
May 13, 2002
MathSciNet review:
1929020
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Abstract | References | Similar Articles | Additional Information
Abstract: If $p$ is an odd prime, $G$ a finite group and $P$ a Sylow-$p$-subgroup of $G$, a theorem of Glauberman and Thompson states that $G$ is $p$-nilpotent if and only if $N_{G}(Z(J(P)))$ is $p$-nilpotent, where $J(P)$ is the Thompson subgroup of $P$ generated by all abelian subgroups of $P$ of maximal order. Following a suggestion of G. R. Robinson, we prove a block-theoretic analogue of this theorem.
- J. Alperin and Michel Broué, Local methods in block theory, Ann. of Math. (2) 110 (1979), no. 1, 143–157. MR 541333, DOI https://doi.org/10.2307/1971248
- Richard Brauer, On the structure of blocks of characters of finite groups, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Springer, Berlin, 1974, pp. 103–130. Lecture Notes in Math., Vol. 372. MR 0352238
- Michel Broué and Lluís Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), no. 2, 117–128. MR 558864, DOI https://doi.org/10.1007/BF01392547
- Marc Cabanes, Extensions of $p$-groups and construction of characters, Comm. Algebra 15 (1987), no. 6, 1297–1311. MR 882955, DOI https://doi.org/10.1080/00927878708823470
- Daniel Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Co., New York, 1980. MR 569209
- Burkhard Külshammer and Lluís Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), no. 1, 17–71. MR 1069239, DOI https://doi.org/10.1007/BF01233419
- 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
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Additional Information
Radha Kessar
Affiliation:
Department of Mathematics, University College, High Street, Oxford OX14BH, United Kingdom
Address at time of publication:
Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210
MR Author ID:
614227
Markus Linckelmann
Affiliation:
CNRS, Université Paris 7, UFR Mathématiques, 2, place Jussieu, 75251 Paris Cedex 05, France
Address at time of publication:
Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210
MR Author ID:
240411
Received by editor(s):
June 14, 2001
Received by editor(s) in revised form:
August 15, 2001
Published electronically:
May 13, 2002
Communicated by:
Stephen D. Smith
Article copyright:
© Copyright 2002
American Mathematical Society