×

Cohomology of infinite groups realizing fusion systems. (English) Zbl 1434.55007

Let \(G\) be a discrete group, \(p\) a prime number and \(S\) a finite \(p\)-group contained in \(G\). The subgroup \(S\) is called a Sylow \(p\)-subgroup if every \(p\)-subgroup of \(G\) is conjugate to a subgroup of \(S\). Given these data, the fusion system, \(\mathcal{F}_{S}(G)\), is the category with objects the subgropus of \(S\) and morphisms given by maps induced by conjugation by elements of \(G\). In general a fusion system \(\mathcal{F}\) is a category with objects the subgroups of a finite \(p\)-group \(S\) and injective homomorphisms satisfying certain axioms. The primary examples are the Sylow \(p\)-subgroups when \(G\) is a finite group. There are many examples of abstract fusion systems that do not arise from finite groups. If \(\mathcal{F}= \mathcal{F}_{S}(G)\) we say that the fusion system is realized by the group \(G\).
Given an abstract fusion system \(\mathcal{F}\) there are at least two ways of constructing a group \(\pi\) that realizes this fusion system, one given by I. J. Leary and R. Stancu [Algebra Number Theory 1, No. 1, 17–34 (2007; Zbl 1131.20012)] and the other one by G. R. Robinson [J. Algebra 314, No. 2, 912–923 (2007; Zbl 1184.20024)]. Let \(G\) be a finite group with associated fusion system \(\mathcal{F}_{S}(G)\) and let \(\pi\) be the group that realizes \(\mathcal{F}_{S}(G)\) (given by Leray-Stancu or Robinson). The authors explore relations between \(G\) and \(\pi\). For example, they prove the following:
Theorem 1.1. Given \(G\) and \(\pi\) as above, there is a short exact sequence of groups \(1\to F\to\pi \to G\to 1\), where \(F\) is a free group and there is an isomorphism \(H^{n}(\pi;\mathbb{F}_{p})\cong H^{n} (G;\mathbb{F}_{p})\oplus H^{n-1}(G;\mathrm{Hom}(F_{ab},\mathbb{F}_{p}))\).
The authors provide infinitely many examples of mod 2 fusion systems where the second cohomology of the group \(\pi\) is not isomorphic to that of \(G\).

MSC:

55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
20J06 Cohomology of groups
20E08 Groups acting on trees
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aschbacher, M., Chermak, A.: A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver. Ann. Math. (2) 171, 881-978 (2010) · Zbl 1213.20017 · doi:10.4007/annals.2010.171.881
[2] Aschbacher, M., Kessar, R., Oliver, B.: Fusion Systems in Algebra and Topology, London Mathematical Society Lecture Note Series, 391. Cambridge University Press, Cambridge (2011) · Zbl 1255.20001 · doi:10.1017/CBO9781139003841
[3] Benson, D.J.: Representations and Cohomology II, Cambridge Studies in Advanced Mathematics, vol. 30. Cambridge University Press, Cambridge (1998) · Zbl 0908.20001
[4] Benson, D.J., Wilkerson, C.W.: Finite simple groups and Dickson invariants. In: Homotopy Teory and its Applications (Cocoyoc, 1993), Contemporary Mathematics, vol. 188,, pp. 31-42 American Mathemtical Society, Providence (1995) · Zbl 0840.55009
[5] Broto, C., Levi, R., Oliver, B.: Homotopy equivalences of \[p\] p-completed classifying spaces of finite groups. Invent. Math. 151, 611-664 (2003) · Zbl 1042.55008 · doi:10.1007/s00222-002-0264-5
[6] Broto, C., Levi, R., Oliver, B.: The homotopy theory of fusion systems. J. Am. Math. Soc. 16, 779-856 (2003) · Zbl 1033.55010 · doi:10.1090/S0894-0347-03-00434-X
[7] Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics. Springer, New York (1982) · Zbl 0584.20036 · doi:10.1007/978-1-4684-9327-6
[8] Cohen, D.E.: Combinatorial Group Theory: A Topological Approach, London Mathemticai Society Student Texts, vol. 14. Cambridge University Press, Cambridge (1989) · Zbl 0697.20001 · doi:10.1017/CBO9780511565878
[9] Craven, D.A.: The Theory of Fusion Systems. Cambridge University Press, Cambridge (2011) · Zbl 1278.20001 · doi:10.1017/CBO9780511794506
[10] Dwyer, W.G., Henn, H.-W.: Homotopy Theoretic Methods in Group Cohomology, Advanced Courses in Mathematics—CRM Barcelona. Birkhauser Verlag, Basel (2001) · Zbl 1047.55001 · doi:10.1007/978-3-0348-8356-6
[11] Goldschmidt, D.M.: A conjugation family for finite groups. J. Algebra 16, 138-142 (1970) · Zbl 0198.04306 · doi:10.1016/0021-8693(70)90046-3
[12] Gorenstein, D., Lyons, R., Solomon, R.: Mathematical Surveys and Monographs, vol. 40. The Classification of the Finite Simple Groups, Number 3. American Mathematical Society, Providence, RI (1994) · Zbl 0816.20016
[13] Leary, I.J., Stancu, R.: Realising fusion systems. Algebra Number Theory 1, 17-34 (2007) · Zbl 1131.20012 · doi:10.2140/ant.2007.1.17
[14] Libman, A., Seeliger, N.: Homology decompositions and groups inducing fusion systems. Homol. Homotopy Appl. 14(2), 167-187 (2012) · Zbl 1267.55006 · doi:10.4310/HHA.2012.v14.n2.a10
[15] Libman, A., Viruel, A.: On the homotopy type of the non-completed classifying space of a \[p\] p-local finite group. Forum Math. 21, 723-757 (2009) · Zbl 1168.55005 · doi:10.1515/FORUM.2009.036
[16] Malle, G., Testerman, D.: Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge (2011) · Zbl 1256.20045 · doi:10.1017/CBO9780511994777
[17] Oliver, B., Ventura, J.: Extensions of linking systems with \[p\] p-group kernel. Math. Ann. 338, 983-1043 (2007) · Zbl 1134.55011 · doi:10.1007/s00208-007-0104-4
[18] Robinson, G.R.: Amalgams, blocks, weights, fusion systems and finite simple groups. J. Algebra 314, 912-923 (2007) · Zbl 1184.20024 · doi:10.1016/j.jalgebra.2007.05.010
[19] Scott, G.P., Wall, C.T.C.: Topological Methods in Group Theory, Homological Group Theory, LMS Lecture Notes 36. Cambridge University Press, Cambridge (1979)
[20] Seeliger, N.: Signalizer functors, existence, and the fundamental group, preprint. arXiv:1105.3403v7
[21] Serre, J.-P.: Trees. Springer Monographs in Mathematics, Translated from French original by John Stillwell. Springer, Berlin (1980) · Zbl 0548.20018
[22] Webb, P.J.: A local method in group cohomology. Comment. Math. Helvetici 62, 135-167 (1987) · Zbl 0616.20022 · doi:10.1007/BF02564442
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.