A note on the automorphism group of the Hamming graph

Let $m>۱$ be an integer and $\Omega$ be an $m$-set‎. ‎The Hamming graph $H(n,m)$ has $\Omega ^{n}$ as its vertex-set‎, ‎with two vertices are adjacent if and only if they differ in exactly one coordinate‎. ‎In this paper‎, ‎we provide a new proof on the automorphism group of the Hamming graph $H(n,m)$‎. ‎Although our result is not new (CE Praeger‎, ‎C Schneider‎, ‎Permutation groups and Cartesian decompositions‎, ‎Cambridge University Press‎, ‎۲۰۱۸)‎, ‎we believe that our proof is shorter and more elementary than the known proofs for determining the automorphism group of Hamming graph‎.