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

The papers during this assortment, all absolutely refereed, unique papers, replicate many elements of contemporary major advances in homotopy thought and crew cohomology. From the Contents: A. Adem: at the geometry and cohomology of finite uncomplicated 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.- ok. Ishiguro: Classifying areas of compact uncomplicated 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 house of the 4-dim unipotent group.- J.E. McClure, L. Smith: at the homotopy area of expertise of BU(2) at the best 2.- G. Mislin: Cohomologically primary parts and fusion in teams.

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4 (1988), no. 4, 372-382. [16] P. HILTON, G. MISLIN and J. ROITBERG, Localization of N{lpotent Groups and Spaces, North-Holland Math. Studies 15 (1975). [17] R. C. LYNDON and P. E. ScItuPP, Combinatorial Group Theory, Ergeb. Math. Grenzgeb. 89, Springer-Verlag, 1977. [18] S. MACLANE, Categories for the Working Mathematician, Graduate Texts in Math. 5, Springer-Verlag, 1975. [19] G. MISLIN, Nilpotent groups with finite commutator subgroup, Lecture Notes in Math. 418, Springer-Verlag, 1974, 103-120.

10 Remark: Let Sqk, k E Z, the operation defind by Sqkx = Sql*l-kx if x is an element of degree [x[ of an A~-module M. We verify that Ek~2kM, k > 0 and M E/4, is the quotient of M by its sub-A~-module generated by Im Sqo + . . + Im Sqk-1. It is proved in [LS] that N is nilpotent if and only if for any x E N, there exists r > 0 such that Sq~x = 0 (Sqo r times). 9 is now equivalent to the following statement. An unstable A~-module is m-nilpotent if and only if for any x E N and for any k: 0 < k < m, there exists rk >_ 0 such that Sq~k*x = 0 (Sqk rk-times) (see iS]).

Direct proofs of these statements can also be given using basic properties of the P-localization functor [20,21,22]. 1 Every homomorphic image of a generically trivial group is generically trivial. 2 If N >--+G ~ Q is a group extension zn wn,cu N and Q are generically trivial, then G is also generically trivial. 3 The (restricted) direct product of a family of generically trivial groups is generically trivial. 4 The free product of a faraily of generically trivial groups is generically trivial Thus, the class of generically trivial groups is closed under small colimits.