We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [ ]2[x1, ;…, xn] = [oplus ]d[ges ]0Pd(n), viewed as a graded left module over the Steenrod algebra [Ascr ] at the prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the n variables x1, …, xn of grading 1. The present article investigates the hit problem for the [Ascr ]-submodule of symmetric polynomials B(n) = P(n)[sum ]n , where [sum ]n denotes the symmetric group on n letters acting on the right of P(n). Among the main results is the symmetric version of the well-known Peterson conjecture. For a positive integer d, let μ(d) denote the smallest value of k for which d = [sum ]ki=1(2λi−1), where λi [ges ] 0.