By Barcelona Conference on Algebraic Topology 1990 San Feliu De Guixols, Manuel Castellet, J. Aguade, Frederick R. Cohen

ISBN-10: 0387551956

ISBN-13: 9780387551951

The papers during this assortment, all absolutely refereed, unique papers, mirror many points of contemporary major advances in homotopy conception and staff cohomology. From the Contents: A. Adem: at the geometry and cohomology of finite basic groups.- D.J. Benson: Resolutions and Poincar duality for finite groups.- C. Broto and S. Zarati: On sub-A*-algebras of H*V.- M.J. Hopkins, N.J. Kuhn, D.C. Ravenel: Morava K-theories of classifying areas and generalized characters for finite groups.- okay. Ishiguro: Classifying areas of compact easy lie teams and p-tori.- A.T. Lundell: Concise tables of James numbers and a few homotopyof classical Lie teams and linked homogeneous spaces.- J.R. Martino: Anexample of a sturdy splitting: the classifying area of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy distinctiveness of BU(2) at the best 2.- G. Mislin: Cohomologically critical parts and fusion in teams.

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**Additional info for Algebraic Topology: Homotopy and Group Cohomology : Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Held in S. Feliu De Guixols, **

**Example text**

RIBENBOIM, Torsion et localisation de groupes arbitraires, Lecture Notes in Math. 740, Springer-Verlag, 1978, 444-456. [21] P. RIBENBOIM, Equations in groups, with special emphasis on localization and torsion I, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 19 (1987), no. 2, 23-60. [22] P. RIBENBOIM, Equations in groups, with special emphasis on localization and torsion II, Portugal. Math. 44 (1987), fasc. 4, 417-445. [23] D. J. S. ROBINSON, Finiteness Conditions and Generalized Soluble Groups, Part P, Ergeb.

6 For a finite group G, the following assertions are equivalent: (a) G is generically triviab (b) G is perfect; (c) for every prime p, G is generated by p'-torsion elements. 23 PROOF. The implications ( c ) ~ ( a ) ~ ( b ) hold for all groups G. If G is finite, then, for each set of primes P, l: G --* Gp is an epimorphism onto a P-group, and Kerl is generated by the set of P'-torsion elements of G; cf. [22, ยง7]. This shows that (a)=~(c). To prove that (b)=t-(a), observe that, given a prime p, Gp is perfect because it is a homomorphic image of G, and also nilpotent because it is a finite p-group.

It also contains all groups which are p-local for some prime p, and it is closed under taking subgroups and forming cartesian products; cf. 10]. Thus it is dosed under small (inverse) limits. 1 Residually nilpotent groups are separable. [] If a group G is not separable, then one cannot expect to recover full information about G from the family of its p-localizations Gp. The worst possible situation occurs, of course, when all these vanish. We introduce new terminology to analyze this case. 2 A group G is called generically trivial if Gp = 1 for all primes p.

### Algebraic Topology: Homotopy and Group Cohomology : Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Held in S. Feliu De Guixols, by Barcelona Conference on Algebraic Topology 1990 San Feliu De Guixols, Manuel Castellet, J. Aguade, Frederick R. Cohen

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