**white@brahms.BERKELEY.EDU (Samuel P. White)** (02/07/86)

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