Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix?
More formally, let $A_1$ be the matrix $$ A_1 = \left[\begin{array}{cccc} 1 & 2 & \cdots & n \\ n+1 & n+2 & \cdots & 2n \\ \vdots & \vdots & \ddots &\vdots \\ (m-1)n+1 & (m-1)n + 2 & \cdots & mn \end{array}\right] $$ and let $A_2$ be an $m\times n$ matrix such that each of the numbers $1,\ldots, mn$ appears (exactly once) as an entry in $A_2$. Is there necessarily a sequence of row and column permutations that transforms $A_1$ into $A_2$? If not, can one easily characterize which permutations are generated by row and column permutations?