Cohomology of groups with metacyclic Sylow \(p\)-subgroups. (English) Zbl 0869.55014
The results in this paper are the determination of the cohomology algebra \(H^*(G; \mathbb{F}_p)\) where \(G\) is a finite group having non-abelian metacyclic \(p\)-Sylow subgroups, and the complete \(p\)-local stable decomposition of the classifying space \(BG\). The first result depends on the result that for a metacyclic \(p\)-group the stable elements coincide with those invariant under the action of the normalizer, the second on earlier work by one of the authors. Note that throughout \(p\) is an odd prime.
Reviewer: C.B.Thomas (Cambridge)
MSC:
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 20J06 | Cohomology of groups |
References:
| [1] | D. J. Benson and M. Feshbach, Stable splittings of classifying spaces of finite groups, Topology 31 (1992), no. 1, 157 – 176. · Zbl 0752.55008 · doi:10.1016/0040-9383(92)90068-S |
| [3] | Jill Dietz, Stable splittings of classifying spaces of metacyclic \?-groups, \? odd, J. Pure Appl. Algebra 90 (1993), no. 2, 115 – 136. · Zbl 0790.55011 · doi:10.1016/0022-4049(93)90125-D |
| [4] | Jill Dietz, Stable splittings of classifying spaces of metacyclic 2-groups, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 2, 285 – 299. · Zbl 0826.55011 · doi:10.1017/S0305004100072583 |
| [5] | D. J. Glover, A study of certain modular representations, J. Algebra 51 (1978), no. 2, 425 – 475. · Zbl 0376.20008 · doi:10.1016/0021-8693(78)90116-3 |
| [6] | John C. Harris and Nicholas J. Kuhn, Stable decompositions of classifying spaces of finite abelian \?-groups, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 3, 427 – 449. · Zbl 0686.55007 · doi:10.1017/S0305004100065038 |
| [7] | Johannes Huebschmann, The mod-\? cohomology rings of metacyclic groups, J. Pure Appl. Algebra 60 (1989), no. 1, 53 – 103. · Zbl 0688.20032 · doi:10.1016/0022-4049(89)90107-2 |
| [8] | B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). · Zbl 0217.07201 |
| [9] | John R. Martino, Classifying spaces of \?-groups with cyclic maximal subgroups, Topology and representation theory (Evanston, IL, 1992) Contemp. Math., vol. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 157 – 174. · Zbl 0830.20077 · doi:10.1090/conm/158/01457 |
| [10] | J. Martino and S. Priddy, On the cohomology and homotopy of Swan groups (to appear). · Zbl 0882.20034 |
| [11] | John Martino and Stewart Priddy, The complete stable splitting for the classifying space of a finite group, Topology 31 (1992), no. 1, 143 – 156. · Zbl 0752.55010 · doi:10.1016/0040-9383(92)90067-R |
| [12] | John Martino and Stewart Priddy, Classification of \?\? for groups with dihedral or quarternion Sylow 2-subgroups, J. Pure Appl. Algebra 73 (1991), no. 1, 13 – 21. · Zbl 0741.55006 · doi:10.1016/0022-4049(91)90103-9 |
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