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> Does there exist an isomorphism between the semigroups $S(4)$ and ‎‎‎‎‎‎$\mathbf Z\_{256‎‎‎‎‎‎‎}$.‎ 
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$S(4)$ is the set of all maps from the set $X$ to itself and $X = \{1, 2, 3, 4\}$. $S(4)$ is a semigroup under the composition of mappings and ‎‎‎‎‎$\mathbf Z\_{256} = {0, 1, 2, … , 255}$ is the semigroup under multiplication of integers modulo 256.