# Semigroup isomorphism

Does there exist an isomorphism between the semigroups $S(4)$ and $\mathbb{‎‎‎‎‎Z}_{‎256}$?‎

$S(4)$ 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 $‎‎‎‎‎‎\mathbb{Z}_{‎256}=\{0, 1, 2, \dots , 255\}$ is the semigroup under multiplication modulo 256.

The semigroup $S(4)$ contains exactly $24$ invertible elements (which form the symmetric group $S_4$). But $256^Z$ contains $128$ invertible elements (all odds). So they can't be isomorphic.

Alternatively: $256^Z$ has an absorbing element $0$ with $0x=x0=0$ for all $x$. No such map exists in $S(4)$.