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By T.Y. Lam

By way of aiming the extent of writing on the beginner instead of the gourmand and through stressing the position of examples and motivation, the writer has produced a textual content that's appropriate for a one-semester graduate direction or for self-study.

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Example text

16 Let G be a finite group. (i) The directed graph (or partially ordered set) va1(G) of finite Gmodules has the isomorphism classes [M] of finite G-modules M as its vertices. ad(G), we have an edge (X, Y), if and only if X = [M] for some finite G-module M with a minimal G-submodule N such that Y = [MIN]. (ii) For a prime number p, we denote the subgraph of vd (G) of all finite G-modules of p-power order (that is, all finite l E G-modules) by &aalp (G). (iii) For any set ic of primes, (G) denotes the corresponding subgraph of G-modules of order divisible only by primes in ic.

Hence the partitions can be used to label the vertices of Wp. The edges are also independent of p; there is in edge from vertex ) to vertex ,u, if and only if there exists a j with ) = µ, for all i Oj and 2; = µj+ 1 . In particular, the number of edges ending at vertex 2 is equal to the number of distinct parts of ), including 0; that is I {)Ii= 1,2, ... } I. This proves the local finiteness and unboundedness of 4ep. pA, The second example is similar as far as the localization procedure is concerned, but the components are infinitely more complicated.

So suppose that n > 0, and let e be the block idempotent with Me = M. Then the injective 30 PERFECT GROUPS WITH NONTRIVIAL FITTING SUBGROUP Zp G-hull I of M also satisfies le =1, since direct summands of injective modules are injective. If X is the cokernel of the injection M --* I, we get, from the long exact sequence of cohomology and the fact that Hr (G, I) = 0 for all r > 0, H" (G, M) H" -1(G, X), and the result follows by inductive hypothesis. 26 (Block decomposition of 4td x # (G)). Assume that G is perfect.

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