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Sparse fusion systems

Part of: Representation theory of groups

Published online by Cambridge University Press:  19 November 2012

Adam Glesser
Adam Glesser*
Affiliation:
Department of Mathematics, California State University Fullerton, Fullerton, CA 92831, USA (algesser@fullerton.edu)
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Abstract

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We define sparse saturated fusion systems and show that, for odd primes, sparse systems are constrained. This simplifies the proof of the Glauberman–Thompson p-Nilpotency Theorem for fusion systems and a related theorem of Stellmacher. We then define a more restrictive class of saturated fusion systems, called extremely sparse systems, that are constrained for all primes.

Keywords

fusion systemsp-nilpotencygroupssparse

MSC classification

Secondary: 20C20: Modular representations and characters 20D15: Nilpotent groups, $p$-groups 20D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 135 - 150
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Alperin, J. L. and Broué, M., Local methods in block theory, Annals Math. 110(1) (1979), 143–157.CrossRefGoogle Scholar
2.Broto, C., Levi, R. and Oliver, B., The homotopy theory of fusion systems, J. Am. Math. Soc. 16(4) (2003), 779–856.CrossRefGoogle Scholar
3.Broto, C., Castellana, N., Grodal, J., Levi, R. and Oliver, B., Subgroup families controlling p-local finite groups, Proc. Lond. Math. Soc. 91 (2005), 325–354.CrossRefGoogle Scholar
4.Broto, C., Castellana, N., Grodal, J., Levi, R. and Oliver, B., Extensions of p-local finite groups, Trans. Am. Math. Soc. 359(8) (2007), 3791–3858.CrossRefGoogle Scholar
5.Broué, M. and Puig, L., Characters and local structure in G-algebras, J. Alg. 63 (1980), 306–317.CrossRefGoogle Scholar
6.Broué, M. and Puig, L., A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.CrossRefGoogle Scholar
7.Craven, D. A., Control of fusion and solubility in fusion systems, J. Alg. 323(9) (2010), 2429–2448.CrossRefGoogle Scholar
8.Díaz, A., Glesser, A., Mazza, N. and Park, S., Glauberman's and Thompson's theorems for fusion systems, Proc. Am. Math. Soc. 137 (2009), 495–503.CrossRefGoogle Scholar
9.Díaz, A., Glesser, A., Mazza, N. and Park, S., Control of transfer and weak closure in fusion systems, J. Alg. 323 (2010), 382–392.CrossRefGoogle Scholar
10.Gorenstein, D., Finite groups, 2nd edn (Chelsea, New York, 1980).Google Scholar
11.Higman, D. G., Focal series in finite groups, Can. J. Math. 5 (1953), 477–497.CrossRefGoogle Scholar
12.Kessar, R., The Solomon system (3) does not occur as fusion system of a 2-block, J. Alg. 296 (2006), 409–425.CrossRefGoogle Scholar
13.Kessar, R., Introduction to block theory, in Group representation theory, pp. 47–77 (EPFL Press, Lausanne, 2007).Google Scholar
14.Kessar, R. and Linckelmann, M., ZJ-theorems for fusion systems, Trans. Am. Math. Soc. 360 (2008), 3093–3206.CrossRefGoogle Scholar
15.Külshammer, B., On the structure of block ideals in group algebras of finite groups, Commun. Alg. 19 (1980), 1867–1872.CrossRefGoogle Scholar
16.Linckelmann, M., Introduction to fusion systems, in Group representation theory, pp. 79–113 (EPFL Press, Lausanne, 2007).Google Scholar
17.Onofrei, S. and Stancu, R., A characteristic subgroup for fusion systems, J. Alg. 322(5) (2009), 1705–1718.CrossRefGoogle Scholar
18.Passman, D., Permutation groups (W. A. Benjamin, New York, 1968).Google Scholar
19.Stellmacher, B., A characteristic subgroup of Σ 4-free groups, Israel J. Math. 94 (1996), 367–379.CrossRefGoogle Scholar
20.Thévenaz, J., G-algebras and modular representation theory, Oxford Mathematical Monographs (Clarendon Press/Oxford University Press, New York, 1995).Google Scholar
21.Thompson, J., Normal p-complements for finite groups, J. Alg. 1 (1964), 43–46.CrossRefGoogle Scholar
22.Weigel, T., Finite p-groups which determine p-nilpotency locally, Hokkaido Math. J. 41(1) (2012), 11–29.CrossRefGoogle Scholar