hw2_913-SS11

# hw2_913-SS11 - Math 913 Spring 2011 – HW 2 Due Wednesday...

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Unformatted text preview: Math 913, Spring 2011 – HW 2 Due Wednesday, 27 April 2011 1. Let m : V × W-→ D be a nondegenerate pairing. For finite dimensional U ≤ V and Y ≤ W , let Y 1 be a complement to ( U ∩ ⊥ Y ) ⊥ in W and let U 1 be a complement to ⊥ ( Y ∩ U ⊥ ) in V . Set U = U ⊕ U 1 and Y = Y ⊕ Y 1 . Prove that the restriction of m to U × Y is nondegenerate with dim U = dim y ≤ 2 max(dim U , dim Y ). 2. (a) Prove that the group G is simple if and only if for every pair of nonidentity elements g and h in G it is possible to write h as the product of a finite number of conjugates of g and g- 1 . (b) State and prove the corresponding result for quasisimple G . (Recall that G is quasisimple provided it is perfect G = G and G/Z ( G ) is simple.) 3. Let { G i | i ∈ I } be a directed system of subgroups in the group G , which is to say that G = S i ∈ I G i and that for every i,j ∈ I there is a k ∈ I with h G i ,G j i ≤ G k . Prove that if each G i is quasisimple, then G is quasisimple.is quasisimple....
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hw2_913-SS11 - Math 913 Spring 2011 – HW 2 Due Wednesday...

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