Let us consider the prime field of two elements, $\mathbb F_2.$ One of the open problems in Algebraic topology is the hit problem for a module over the mod 2 Steenrod algebra $\mathscr A$. More specifically, this problem asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$ regarded as a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots, x_m,$ each of degree one. The algebra $\mathcal P_m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan space of type $(\mathbb F_2, 1).$ The hit problem has been thoroughly studied for 35 years in a variety of contexts by many authors and completely solved for $m\leq 4.$ Furthermore, it has been closely related to some classical problems in the homotopy theory and applied in studying the $m$-th Singer algebraic transfer $Tr^{\mathscr A}_m$ \cite{W.S1}. This transfer is one of the useful tools for studying the Adams $E^{2}$-term, ${\rm Ext}_{\mathscr A}^{*, *}(\mathbb F_2, \mathbb F_2) = H^{*, *}(\mathscr A, \mathbb F_2).$The aim of this work is to continue our study of the hit problem of five variables. At the same time, this result will be applied to the investigation of the fifth transfer of Singer and the modular representation of the general linear group of rank 5 over $\mathbb F_2.$ More precisely, we grew out of a previous result of us in \cite{D.P3} on the hit problem for $\mathscr A$-module $\mathcal P_5$ in the generic degree $5(2^t-1) + 18.2^t$ with $t$ an arbitrary non-negative integer. The result confirms Sum's conjecture \cite{N.S2} on the relation between the minimal set of $\mathscr A$-generators for the polynomial algebras $\mathcal P_{m-1}$ and $\mathcal P_{m}$ in the case $m=5$ and the above generic degree. Moreover, by using our result \cite{D.P3} and a presentation in the $\lambda$-algebra of $Tr_5^{\mathscr A}$, we show that the non-trivial element $h_1e_0 = h_0f_0\in {\rm Ext}_{\mathscr A}^{5, 5+(5(2^0-1) + 18.2^0)}(\mathbb F_2, \mathbb F_2)$ is in the image of the fifth transfer and that $Tr^{\mathscr A}_5$ is an isomorphism in the bidegree $(5, 5+(5(2^0-1) + 18.2^0)).$ In addition, the behavior of $Tr^{\mathscr A}_5$ in the bidegree $(5, 5+(5(2^t-1) + 18.2^t))$ when $t\geq 1$ was also discussed. This method is different from that of Singer in studying the image of the algebraic transfer.
Balanced finite presentations of the trivial group
