Tokyo Journal of Mathematics

On the Reduced Lefschetz Module and the Centric $p$-Radical Subgroups


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The purpose of this paper is to show that the reduced Lefschetz module of the $G$-poset $\mathcal{B}_{p}^{cen}(G)$ consisting of all centric $p$-radical subgroups of a finite group $G$ is an $\mathcal{X}$-projective virtual $\mathbb{Z}_{p}[G]$-module where $\mathcal{X}$ is a family of $p$-subgroups of the normalizers of non-centric $p$-radical subgroups of $G$. As corollary, we have a lower bound of the $p$-power of the reduced Euler characteristic $\tilde{\chi}(\mathcal{B}_{p}^{cen}(G))$.

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Tokyo J. Math., Volume 28, Number 1 (2005), 79-90.

First available in Project Euclid: 5 June 2009

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SAWABE, Masato. On the Reduced Lefschetz Module and the Centric $p$-Radical Subgroups. Tokyo J. Math. 28 (2005), no. 1, 79--90. doi:10.3836/tjm/1244208281.

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  • D. J. Benson, Representations and cohomology II, Cambridge studies in advanced mathematics 31 (1991), Cambridge University Press.
  • K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87 (1982), Springer.
  • S. B. Conlon, Decompositions induced from the Burnside algebra, J. Algebra 10 (1968), 102–122.
  • W. G. Dwyer, Sharp homology decompositions for classifying spaces of finite groups, Group Representations: Cohomology, Group Actions and Topology, Proc. Sympos. Pure Math. 63 (1998). Amer. Math. Soc., 197–220,
  • D. Gluck, Idempotent formula for the Burnside algebra with applications to the $p$-subgroups simplicial complex, Illinois J. Math. 25 (1981), 63–67.
  • J. Grodal, Higher limits via subgroup complexes, Ann. of Math. 155 (2002), 405–457.
  • D. Quillen, Homotopy properties of the poset of nontrivial $p$-subgroups of a group, Adv. Math. 28 (1978), 101–128.
  • M. Sawabe, $2$-radical subgroups of the Conway simple group $Co_{1}$, J. Algebra 211 (1999), 115–133.
  • M. Sawabe, The centric $p$-radical complex and a related $p$-local geometry, Math. Proc. Cambridge Philos. Soc. 133 (2002), 383–398.
  • S. D. Smith, Constructing representations from group geometries, Arcata Conference on Representations of Finite Groups, Proc. Sympos. Pure Math. 47 (1987), Amer. Math. Soc., 303–313.
  • S. D. Smith and S. Yoshiara, Some homotopy equivalences for sporadic geometries, J. Algebra 192 (1997), 326–379.
  • J. Thévenaz, Permutation representations arising from simplicial complexes, J. Combinatorial Th., Ser. A 46 (1987), 121–155.
  • J. Thévenaz and P. Webb, Homotopy equivalence of posets with a group action, J. Combinatorial Th., Ser. A 56 (1991), 173–181.
  • P. Webb, A local method in group cohomology, Comment. Math. Helv. 62 (1987), 135–167.
  • S. Yoshiara, Radical $2$-subgroups of the Monster and Baby Monster, preprint.
  • T. Yoshida, Idempotents in Burnside ring and Dress induction theorem, J. Algebra 80 (1983), 90–105.