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Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbbZ_2.$ It is well-known that the classical "hit problem" for a module over themod 2 Steenrod algebra $\mathscr A$ is an interesting and important openproblem of Algebraic topology, which asks a minimal set of generators for thepolynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, regardedas a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots,x_m,$ each of degree 1. The algebra $\mathcal P_m$ is the $\mathbbF_2$-cohomology of the product of $m$ copies of the Eilenberg-MacLan complex$K(\mathbb F_2, 1).$ Although the hit problem has been thoroughly studied formore than 3 decades, solving it remains a mystery for $m\geq 5.$ Our intent inthis work is of studying the hit problem of five variables. More precisely, wedevelop our previous work [Commun. Korean Math. Soc. 35 (2020), 371-399] on thehit problem for $\mathscr A$-module $\mathcal P_5$ in a degree of the genericform $n_t:=5(2^t-1) + 18.2^t,$ for any non-negative integer $t.$ An efficientapproach to solve this problem had been presented. Two applications of thisstudy are to determine the dimension of $\mathcal P_6$ in the generic degree$5(2^{t+4}-1) + n_1.2^{t+4}$ for all $t > 0$ and to describe the modularrepresentations of the general linear group of rank 5 over $\mathbb F_2.$ As acorollary, the cohomological "transfer", defined by William Singer [Math. Z.202 (1989), 493-523], is an isomorphism in bidegree $(5, 5+n_0).$ Singer'stransfer is one of the relatively efficient tools to approach the structure ofmod-2 cohomology of the Steenrod algebra.

On modules over the mod 2 Steenrod algebra and hit problems