From: Robin Chapman Subject: Nilpotent elements in group algebras Date: Thu, 15 Jul 1999 09:07:41 GMT Newsgroups: sci.math Here's a nice little result: Let K be a field of characteristic zero, and let G be a finite group. Consider the group algebra A = KG. If x = sum_g a_g g in A is nilpotent then a_1 = 0. Here's a proof. Consider the regular representation. In particular consider the K-endomorphism phi of A given by left multiplication by x, so that phi(y) = x(y). Then if x is nilpotent, so is phi, and the trace of phi vanishes. But this trace is |G|a_1, so a_1 = 0. As we might expect the result holds if K has finite characteristic coprime to |G|. (But certainly not if its characteristic divides |G|.) Questions: (i) Is there an "elementary" proof, in the sense of not using (even the modest amount used above) representation theory. (ii) Same as (i) in the special case where x^2 = 0. (iii) Is the result true when G is an infinite group? -- Robin Chapman http://www.maths.ex.ac.uk/~rjc/rjc.html "They did not have proper palms at home in Exeter." Peter Carey, _Oscar and Lucinda_ Sent via Deja.com http://www.deja.com/ Share what you know. Learn what you don't. ============================================================================== From: wcw@math.psu.edu (William C Waterhouse) Subject: Re: Nilpotent elements in group algebras Date: 16 Jul 1999 21:18:23 GMT Newsgroups: sci.math In article <7mk8co$at8$1@nnrp1.deja.com>, Robin Chapman writes: > Here's a nice little result: > > Let K be a field of characteristic zero, and let G be a finite group. > Consider the group algebra A = KG. If x = sum_g a_g g in A > is nilpotent then a_1 = 0. >... > (iii) Is the result true when G is an infinite group? Yes, there's a proof on p. 47 of D.S. Passman's book _The Algebraic Structure of Group Rings_. It falls into two parts: 1) In characteristic p, show that nilpotence forces a_1 + \sum {a_g | g has p-power order} = 0. 2) Suppose now we have a nilpotent element in characteristic 0 with a_1 nonzero. Consider valuations on K where all the nonzero a_g are in the valuation ring and have nonzero images in the residue field. Such valuations will occur with residue fields of infinitely many different prime characteristics. But by 1), that forces a_g to be nonzero for g of infinitely many different prime power orders, which is impossible. William C. Waterhouse Penn State