Abstract
The notion of adequate subgroups was introduced by Thorne (J Inst Math Jussieu. arXiv:1107.5989, to appear). It is a weakening of the notion of big subgroup used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to prove some new lifting theorems. It was shown in Guralnick et al. (J Inst Math Jussieu. arXiv:1107.5993, to appear) that certain groups were adequate. One of the key aspects was the question of whether the span of the semisimple elements in the group is the full endomorphism ring of an absolutely irreducible module. We show that this is the case in prime characteristic p for p-solvable groups as long the dimension is not divisible by p. We also observe that the condition holds for certain infinite groups. Finally, we present the first examples showing that this condition need not hold and give a negative answer to a question of Richard Taylor.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)
Curtis C.W., Reiner I.: Representation theory of finite groups and associative algebras. Pure and Applied Mathematics, vol. XI. Interscience Publishers, Wiley, New York, London (1962)
Fong P.: Some properties of characters of finite solvable groups. Bull. Am. Math. Soc. 66, 116–117 (1960)
Goldstein, D., Guralnick, R.: Cosets of Sylow subgroups and a question of Richard Taylor (preprint)
Guralnick R.: Small representations are completely reducible. J. Algebra 220, 531–541 (1999)
Guralnick, R., Herzig, F., Taylor, R., Thorne, J.: Adequate subgroups. J. Inst. Math. Jussieu. arXiv:1107.5993 (to appear)
Guralnick, R., Malle, G.: Simple groups admit Beauville structures. J. Lond. Math. Soc. arXiv: 1009.6183 (to appear)
Isaacs I.M.: Characters of π-separable groups. J. Algebra 86, 98–128 (1984)
Jantzen, J.: Representations of algebraic groups. 2nd edition. Mathematical Surveys and Monographs, vol. 107. American Mathematical Society, Providence, RI (2003)
Serre J.-P.: Sur la semi-simplicité des produits tensoriels de représentations de groupes. Invent. Math. 116, 513–530 (1994)
Thompson J.G.: On a question of L. J. Paige. Math. Zeitschr. 99, 26–27 (1967)
Thorne, J.: On the automorphy of l-adic Galois representations with small residual image. J. Inst. Math. Jussieu. arXiv:1107.5989 (to appear)
Acknowledgments
The author was partially supported by the NSF Grant DMS-1001962. Some of this work was carried out at the Institute for Advanced Study. The author would like to thank Inna Capdeboscq for her help on constructing the first examples which were not weakly adequate, John Thompson for pointing out his article [11], Simon Guest and Daniel Goldstein for help with some MAGMA computations and Richard Taylor for many helpful comments and questions. Finally, we would like to thank the referee for their careful reading and helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by E. Zelmanov.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Guralnick, R. Adequate subgroups II. Bull. Math. Sci. 2, 193–203 (2012). https://doi.org/10.1007/s13373-011-0018-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13373-011-0018-z