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From: white@brahms.BERKELEY.EDU (Samuel P. White)
Newsgroups: net.math
Subject: Algebraic problems
Message-ID: <11732@ucbvax.BERKELEY.EDU>
Date: Fri, 7-Feb-86 02:26:18 EST
Article-I.D.: ucbvax.11732
Posted: Fri Feb  7 02:26:18 1986
Date-Received: Sun, 9-Feb-86 05:52:13 EST
Sender: usenet@ucbvax.BERKELEY.EDU
Reply-To: white@brahms.BERKELEY.EDU (Samuel P. White)
Organization: University of California, Berkeley
Lines: 39
Keywords: Abelian varieties.  Groups.


      I have a couple of problems I have been working on.  If you
have anything useful to say on either I would be quite interested.

      Let M(n) be the nxn matrices over a field k.  (I do not
think it matters if you assume k is the complex numbers but
I have been told it does not hurt to generalize.)  Let V be
a k-vector subspace which consists entirely of nilpotent elements
of M(n).  Can V generate all of M(n) as an algebra?

      The next problem is a little more complicated, but if you
object to the problem because it is technical and demand
a reason for its existence, the problem does have an intimate
connection with the rank of an abelian variety.  Let G be
a group that acts on the set X = { 1, 2,..., n }, i.e. G is a
subgroup of the permutation group on n letters.  Assume G acts
transitively on the set X and let V be an n-dimensional vector
space over the rationals where G gets a natural representation
on V by permuting the basis elements of V in accordance with
G's action on X.  Does there exist a subset S of V with the
following properties:

        1.)  S is closed under the action of G.
        2.)  S consists of elements whose coordinates are 0's and 1's.
        3.)  There do not exist two disjoint subsets A and B of X
             satisfying the following two properties:
             
             a.)  A and B have the same number of elements.
             b.)  The number of 1's occuring at the coordinates
                  of V listed in A is the same as the number of
                  1's occuring at the coordinates corresponding
                  to the elements of B for every x in S.  In other
                  words, A and B are disjoint subsets of the coordinates
                  of V and I am counting the number of 1's that
                  occur in each subset and seeing if they are equal.
        4.)  S does not span V.   


ucbvax!brahms!white	Samuel P. White/UCB Math Dept/Berkeley CA 94720