Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T02:29:55.335Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of G-posets and Quillen's complex

Part of: Finite geometry and special incidence structures Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Yoav Segev  and
Peter Webb
Yoav Segev
Affiliation:
Department of Mathematics, Ben-Gurion Univesity, Beer-Sheva 84105, Israel
Peter Webb
Affiliation:
Department of Mathematics, University of MinnesotaMinneapolis, Minnesota 55455, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We develop techniques to compute the homology of Quillen's complex of elementary abelian p-subgroups of a finite group in the case where the group has a normal subgroup of order divisible by p. The main result is a long exact sequence relating the homologies of these complexes for the whole group, the normal subgroup, and certain centralizer subgroups. The proof takes place at the level of partially-ordered sets. Notions of suspension and wedge product are considered in this context, which are analogous to the corresponding notions for topological spaces. We conclude with a formula for the generalized Steinberg module of a group with a normal subgroup, and give some examples.

MSC classification

Secondary: 20D30: Series and lattices of subgroups 20C20: Modular representations and characters 51E25: Other finite nonlinear geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 1 , August 1994 , pp. 60 - 75
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Alperin, J. L., ‘A Lie approach to finite groups’, in Groups—Canberra 1989 (ed. Kovács, L. G.), Lecture Notes in Math. 1456 (Springer, Berlin, 1990) pp. 1–9.Google Scholar
[2]Aschbacher, M., and Segev, Y., Locally connected simplical maps’, Israel J. Math 77 (1992), 285–303.CrossRefGoogle Scholar
[3]Bouc, S., ‘Exponentielle et modules de Steinberg’, preprint.Google Scholar
[4]Quillen, D., ‘Homotopy properties of the poset of nontrivial p-subgroups of a group’, Adv. in Math. 28 (1978), 101–128.CrossRefGoogle Scholar
[5]Thévenaz, J., ‘Permutation representations arising from simplical complexes’, J. Combin. Theory Ser. A 46 (1987), 121–155.Google Scholar
[6]Thévenaz, J. and Webb, P. J., ‘Homotopy equivalence of posets with a group action’, J. Combin. Theory Sey. A 56 (1991), 173–181.CrossRefGoogle Scholar
[7]Webb, P. J., ‘Subgroups complexes’, in The Arcata conference on representations of finite groups (ed. Fong, P.),Google Scholar
Proceedings of Symposia in Pure Mathematics vol. 47 (American Mathematical Society, Providence, R. I., 1987) pp. 349–365.Google Scholar