On essential cohomology of powerful p-groups
On essential cohomology of powerful p-groups
For an odd prime p, we prove that a finite powerful p-group having rank two Frattini quotient has nonzeroessential cohomology. We also provide some examples and applications.
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- [1] Adams JF. Problem session for homotopy theory. In: Carlsson G, Cohen R, Miller H, Ravenel D, editorss. Algebraic
Topology, Proceedings of an International Conference; July 27-August 2 1986; Arcata, California. Berlin, Germany:
Springer-Verlag, 1989. pp. 438-443.
- [2] Adem A, Carlson JF, Karagueuzian D, Milgram RJ. The cohomology of the Sylow 2-subgroup of Higman-Sims. J
Pure Appl Algebra 2001; 164: 275-305.
- [3] Adem A, Karagueuzian D. Essential cohomology of finite groups. Comment Math Helv 1997; 72: 101-109.
- [4] Altunbulak Aksu F. Essential cohomology and relative cohomology of finite groups. PhD, Bilkent University, Ankara,
Turkey, 2009.
- [5] Altunbulak Aksu F, Green DJ. Essential cohomology of elementary abelian p-groups. J Pure Appl Algebra 2009;
213: 2238-2243.
- [6] Carlson JF. Depth and transfer maps in the cohomology of groups Math Z 1995; 218: 461-468.
- [7] Carlson JF. Calculating group cohomology: tests for completion. J Symb Comput 2001; 31: 229-242.
- [8] Carlson JF, Townsley L, Valeri-Elizondo L, Zhang M. Cohomology Rings of Finite Groups. With an Appendix:
Calculations of Cohomology Rings of Groups of Order Dividing 64 by Carlson, Valeri-Elizondo and Zhang. Algebras
and Applications. Dordrecht, the Netherlands: Kluwer Academic Publishers, 2003.
- [9] Dixon DJ, Du Sautoy MPF, Mann A, Segal D. Analytic pro-p-groups. Cambridge Studies in Advanced Mathematics.
Cambridge, UK: Cambridge University Press, 1999.
- [10] Green DJ. The essential ideal in group cohomology does not square to zero. J Pure Appl Algebra 2004; 193: 129-139.
- [11] Lubotzky A, Mann A. Powerful p-groups. I. Finite groups. J Algebra 1987; 105: 484-505.
- [12] Mann A, Posnick-Fradkin F. Subgroups of powerful p-groups. Israel J Math 2003; 138: 19-28
- [13] Marx T. The restriction map in cohomology of finite 2-groups. J Pure Appl Algebra 1990; 67: 33-37.
- [14] Minh PA. Essential cohomology and extraspecial p-groups. T Am Math Soc 2001; 353: 1937-1957.
- [15] Quillen D. The spectrum of an equivariant cohomology ring I, II. Ann Math 1971; 94: 549-602.
- [16] Serre JP. Une relation dans la cohomologie des p-groupes. C R Acad Sci Paris 1987; 304: 587-590 (in French).
- [17] Symonds P, Weigel T. Cohomology of p-adic analytic groups. In: du Sautoy M, Segal D, Shalev A, editors. New
Horizons in Pro-p Groups. Progress of Mathematics, Vol. 184. Boston, MA, USA: Birkhäuser, 2000. pp. 349-410.
- [18] Weiss E. Cohomology of Groups. New York, NY, USA: Academic Press, 1969.
- [19] Yalçın E. A note on Serre’s theorem in group cohomology. P Am Math Soc 2008; 136: 2655-2663.