By C. B. Thomas

The aim of this booklet is to review the relation among the illustration ring of a finite workforce and its necessary cohomology by way of attribute periods. during this manner it's attainable to increase the identified calculations and end up a few normal effects for the imperative cohomology ring of a bunch G of best energy order. one of the teams thought of are these of p-rank lower than three, extra-special p-groups, symmetric teams and linear teams over finite fields. a tremendous instrument is the Riemann - Roch formulation which supplies a relation among the attribute periods of an brought on illustration, the periods of the underlying illustration and people of the permutation illustration of the countless symmetric workforce. Dr Thomas additionally discusses the consequences of his paintings for a few mathematics teams as a way to curiosity algebraic quantity theorists. Dr Thomas assumes the reader has taken uncomplicated classes in algebraic topology, workforce concept and homological algebra, yet has incorporated an appendix during which he provides a in simple terms topological evidence of the Riemann - Roch formulation.

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Let A(·) : Po (G) −→ Po (E(G)) be defined by A(·) = Hom(G, ·) ⊗End(G) End(G)/N (End(G)). 1. A(·) is an additive full functor that preserves direct sums. That is, given an E(G)-module map f : A(H) → A(H ) there is a group map φ : H → H such that A(φ) = f . 2. A(·) induces a bijection α : {(H) H ∈ Po (G)} −→ {(P ) P ∈ Po (E(G))} between the set of isomorphism classes (H) of H ∈ Po (G) and the set of isomorphism classes (P ) of P ∈ Po (A(G)). 3. A(·) induces a bijection between the set of quasi-isomorphism classes [H] of H ∈ Po (G) and the set of quasi-isomorphism classes [P ] of finitely generated projective right E(G)-modules.

We say that G has a unique decomposition if 1. G has an indecomposable decomposition G ∼ = G1 ⊕ · · · ⊕ Gt and 2. Given an indecomposable decomposition G ∼ = G1 ⊕ · · · ⊕ Gs then s = t and after a permutation of the subscripts, Gi ∼ = Gi for each i = 1, . . , t. In this case we call G1 ⊕ · · · ⊕ Gt the unique decomposition of G. The unique decomposition of an rtffr group G is necessarily indecomposable. Rtffr groups having unique decomposition are considered to be rare. 4 1. The Fundamental Theorem of Abelian Groups states that if p ∈ Z is a prime and if G is a finite p-group then G has a unique decomposition G = Z/pn1 Z ⊕ · · · ⊕ Z/pnt Z for some integers 0 < n1 ≤ · · · ≤ nt .

H ⊕K ∼ 3. If G(n) = = H ⊕ K for some integer n > 0 and some rtffr groups K and K then K ∼ =K. Proof: We begin the proof with some general comments about finitely generated projective right E(G)-modules. 4 E(G) = E(G1 ) × · · · × E(Gt ) where E(Gi ) = A(Gi ) is indecomposable for each i = 1, . . , t. 34 CHAPTER 2. MOTIVATION BY EXAMPLE Furthermore since E(Gi ) is a pid, given a finitely generated projective right E(G)-module P there are integers p1 , . . , pt ≥ 0 such that P ∼ = E(G1 )(p1 ) ⊕ · · · ⊕ E(Gt )(pt ) .