By Finite Groups 200, Chat-yin Ho

ISBN-10: 3110174472

ISBN-13: 9783110174472

ISBN-10: 3110198126

ISBN-13: 9783110198126

This can be a quantity of study articles relating to finite teams. themes lined comprise the category of finite basic teams, the speculation of p-groups, cohomology of teams, illustration idea and the speculation of structures and geometries. in addition to greater than twenty unique papers at the newest advancements, that allows you to be of serious curiosity to experts, the quantity comprises a number of expository articles, from which scholars and non-experts can find out about the current nation of information and promising instructions for additional study. The Finite teams 2003 convention was once held in honor of John Thompson. The profound effect of his basic contributions is obviously seen during this choice of papers devoted to him.

**Read Online or Download Finite Groups 2003: Proceedings Of The Gainesville Conference On Finite Groups, March 6 - 12, 2003 (De Gruyter Proceedings in Mathematics) PDF**

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**Extra resources for Finite Groups 2003: Proceedings Of The Gainesville Conference On Finite Groups, March 6 - 12, 2003 (De Gruyter Proceedings in Mathematics)**

**Sample text**

Let Zi (P ) be the ith term in the ascending central series; that is Z0 (P ) = 1 and for i > 1, Zi (P ) is the preimage in P of Z(P /Zi−1 (P )). Let Li (P ) be the ith term in the descending central series; that is L0 (P ) = P and for i > 0, Li (P ) = [Li−1 (P ), P ]. 1. Li (P ) ≤ Zc−i (P ). Proof. 6 in [2]. 2. [Li (P ), Zj (P )] ≤ Zj −i−1 (P ), where Zr (P ) = 1 for r ≤ 0. Proof. Let P ∗ = P /Zj −i−1 (P ). If i + 1 ≤ j then Zj (P )∗ = Zi+1 (P ∗ ) and Zj −i−1 (P )∗ = 1, while if i +1 ≥ j then P = P ∗ and Zj −i−1 (P ) = 1.

One obtains the following result. Theorem. Let p ≥ 3(h − 1) and λ ∈ X1 (T ). If either (a) G does not have underlying root system of type Cn (n ≥ 1) or (b) λ, αn∨ = p−2−c 2 , where αn is the unique long simple root and c is odd with 0 < |c| ≤ h − 1, then Ext 1Gσ (Fp ) (L(λ), L(λ)) = 0. Proof. 9 one has Ext1Gσ (Fp ) (L(λ), L(µ)) ∼ = ν∈ Ext 1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))), h X(T )+ | ν, α0∨ where h = {ν ∈ < h}. The only complication in the quasi-split case versus the split case is that the weights ν and σ (ν) may be distinct.

R= ν∈ h By [BNP3, Thm. 4a], the remainder term R is exactly Ext 1G (L(λˆ ), L(µ)). 3. Various conjectures have been made about the dimensions of Ext1 -groups. Here it is shown that in most cases the dimensions of the Ext 1 groups between simple modules for the finite groups are bounded by the dimensions of Ext1 groups for the reductive algebraic groups. The corollary below is a generalization of [BNP3, Thm. 3]. The proof follows along the same lines and details are left to the reader. Corollary.

### Finite Groups 2003: Proceedings Of The Gainesville Conference On Finite Groups, March 6 - 12, 2003 (De Gruyter Proceedings in Mathematics) by Finite Groups 200, Chat-yin Ho

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