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The cohomology of groups has, since its beginnings in the 1920s and 1930s, been the stage for significant interaction between algebra and topology and has led to the creation of important new fields in mathematics, like homological algebra and algebraic K-theory. This is the first book to deal comprehensively with the cohomology of finite groups: it introduces the most important and useful algebraic...
There are intimate connections between the homology and cohomology of the symmetric groups and algebraic topology. The first of these is their connection with the structure of cohomology operations. This arises through Steenrod’s definition of the Pth power operations in terms of properties of certain elements in the groups H* (Sp...
Let G be a finite group and X a space on which G acts. In this chapter we will describe a cohomological analysis of X which involves H*(G; $${\Bbb F}$$ p) in a fundamental way. First developed by Borel and then by Quillen, this approach is the natural generalization of classical Smith Theory. After reviewing the basic constructions and a few examples, we will apply these techniques to certain...
Let G be a finite group. As we have seen, the classifying space BG has a very simple homotopy type as it is a K(G, 1). If G is perfect then H1(G; ℤ) = 0; suppose that we attach cells to BG to obtain a new, but simply-connected complex BG+ with the same homology as before. Or equivalently so that the homotopy fiber of BG→BG+ is acyclic, i. e. Hi; (ℱ ℤ;) = 0 for all...
One of the most remarkable results of this century in mathematics has been the classification — completed in 1980 — of all the finite simple groups. This took over 20 years and occupies almost 5000 pages in the literature, and it is conceivable that there are some errors there, so the details of classification are not really available to us, but the main results can be summarized. There are 17 families...
In this chapter we will describe progress towards understanding the cohomology of the sporadic simple groups. Briefly we recall that from the classification of finite simple groups, [Gor], it was shown that there exist 26 simple groups not belonging to infinite families (i. e. not of alternating or Lie type) and we study ten of these groups here: four of the five Mathieu groups; the Janko groups J...
In this final chapter we apply the techniques of group cohomology to the representation theory of finite groups.Given G a finite group we know that F (G) is semi-simple for any field of characteristic zero.
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