The squaring operartion and the Singer algebraic transfer

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$, with the degree of each $x_i$ being 1, regarded as a module over the mod-2 Steenrod algebra $\mathcal A$, and let $GL_k$ be the general linear group over the prime field $\mathbb F_2$ which acts regularly on $P_k$. We study the algebraic transfer constructed by Singer using the technique of the hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra, $\text{Tor}^{\mathcal A}_{k,k+n} (\mathbb F_2,\mathbb F_2)$, to the subspace of $\mathbb F_2{\otimes}_{\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $n$. In this paper, we extend a result of Hung on the relation between the Singer algebraic transfer and the squaring operation on the cohomology of the Steenrod algebra. Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case $k=5$ and the degree $5(2^{s} -1)$ with $s$ an arbitrary positive integer.