### THE HIT PROBLEM AND THE ALGEBRAIC TRANSFER IN SOME DEGREES

#### Abstract

Denote by Pk the graded polynomial algebra F2[x1, x2, . . . , xk], with the degree of each generator xi being 1, and let GLk be the general linear group over the prime field F2 of two elements which acts naturally on Pk by matrix substitution.

We study the Peterson hit problem of determining a minimal set of generators for Pk as a module over the mod-2 Steenrod algebra, A. In this paper, we study the hit problem in terms of the admissible monomials at the degree (k k 1)(2d − 1). These results are used to verify Singer’s conjecture for the algebraic transfer, which is a homomorphism from the homology of the mod-2 Steenrod algebra, TorAk,k+n(F2, F2), to the subspace of F2⊗APk consisting of all the GLk-invariant classes of degree n. More precisely, using the results on the hit problem, we prove that Singer’s conjecture for the algebraic transfer is true in the case k = 5 and

the degree 4(2d − 1) with d an arbitrary positive integer.

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