Determination of the fifth Singer algebraic transfer in some degrees

Let Pₖ be the graded polynomial algebra F₂[x₁,x₂,… ,xₖ] over the prime field F₂ with two elements and the degree of each variable xᵢ being 1, and let GLₖ be the general linear group over F₂ which acts on Pₖ as the usual manner. The algebra Pₖ is considered as a module over the mod-2 Steenrod algebra A. In 1989, Singer [22] defined the k-th homological algebraic transfer, which is a homomorphism φₖ : TorAk,k+d ( F₂, F₂) → ( F₂⊗APₖ)dGLₖ from the homological group of the mod-2 Steenrod algebra TorAk,k+d ( F₂, F₂) to the subspace ( F₂⊗APₖ)dGLₖ of F₂⊗APₖ consisting of all the GLₖ-invariant classes of degree d. In this paper, by using the results of the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer of rank five is an isomorphism in the internal degrees $d= 20$ and $d = 30$. Our result refutes the proof for the case of $d=20$ in Ph\'uc [17].