Sorting numbers in a matrix by moving an empty entry through other entries is not always possible .

let $\mathbf{A}$ be the set of all $n\times n$ matrices on $\mathbb{N}_{n^2}=\{1,2,...,n^2\}$.

let $T$ be the set of all permutation of $\mathbf{A}$ which swap two adjacent entries. adjacent means one is above/below/right/left of the other (and both are neighbors).

Let $S$ be the permutation subgroup generated by $T$.

Show that $S$ acts intransitively on $\mathbf{A}$.

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