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In this note we shall prove that the Stone-\v{C}ech compactification of$\mathcal{L}^n$ is the space $\bar{\mathcal{L}}^n$ where $\bar{\mathcal{L}}$ isthe extended long line, namely, $\mathcal{L}$ together with its ends $\pm\Omega$. We give a similar description for the Stone-\v{C}ech compactificationof the cartesian power of the semi-closed half-long line $\mathcal{L}_+$. As anapplication we show that any torsion subgroup of the group of allhomeomorphisms of $\cj^n$ (resp. $\cl^n$) is isomorphic to a subgroup of thesymmetric group $S_n$ (resp. the semidirect product $(\bz/2\bz)^n\ltimes S_n$).

Stone-\v{C}ech compactifications and homeomorphisms of products of the long line