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On the homotopy type of the p-subgroup complex for finite solvable groups

Part of: Homotopy theory Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Jürgen Pulkus  and
Volkmar Welker
Jürgen Pulkus
Affiliation:
Mathematisches Institut Universität ErlangenBismarckstr. 1 1/2 91054 ErlangenGermany e-mail: pulkus@mi.uni-erlangen.de
Volkmar Welker
Affiliation:
Fachbereich Mathematik und Informatik Philipps-Universität Marburg35032 MarburgGermany e-mail: welker@mathematik.uni-marburg.de
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Abstract

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We provide a wedge decomposition of the homotopy type of the p-subgroup complex in the case of a finite solvable group G. In particular, this includes a new proof of the result of Quillen which says that this complex is contractible if and only if there is a non-trivial normal p-subgroup in G. We also provide reduction formulas for the G-module structure of the homology groups. Our results are obtained with diagram-methods by gluing the p-subgroup complex of G along the p-subgroup complex of = G/N for a normal p′-subgroup of G.

Keywords

p-subgroup complexhomotopy type of posetsgroup theoryhomotopy co-limits

MSC classification

Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups 20F17: Formations of groups, Fitting classes 55P10: Homotopy equivalences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 2 , October 2000 , pp. 212 - 228
Copyright
Copyright © Australian Mathematical Society 2000

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