Let GL(n, p) denote the general linear group over the prime field Fp, and let Sd denote the right GL(n, p)-module consisting of all homogeneous polynomials of degree d in n variables x1, x2, . . . . x, over F,. We wish to determine the simple composition factors of Sd. Unfortunately, satisfactory models for all the simple GL(n, p) modules themselves are not known. There are p”‘(p 1) of these simple modules, up to isomorphism, and all of them are known to occur in the symmetric algebra S= CF Sd. The general problem is therefore a very difficult one in our present state of knowledge. The purpose of this paper is to offer a solution to this problem in the case n = 3. This extends the work of David Carlisle, who calculated in his Manchester thesis [2] the occurrences of all the l-dimensional modules, and developed a method for dealing with the general case. Let us begin with some general remarks about the problem. The case n = 1 is trivial, as S d= detd, the dth power of the determinant representation. The case n = 2 is also easily disposed of, for instance, by Glover [9]; the simple modules are the tensor products of the Sd for d 1 and d > p the module S, is not simple. For example, the pth power Sf of the natural module is a proper submodule of 411 0021-8693192 $3.00