Extensions of theorems of Gasch\"utz, \v{Z}mud${}^{\mathrm{\prime }}$ and Rhodes on faithful representations

Gasch"utz (1954) proved that a finite group $G$ has a faithful irreducible complex representation if and only if its socle is generated by a single element as a normal subgroup; this result extends to arbitrary fields of characteristic $p$ so long as $G$ has no nontrivial normal $p$-subgroup. \v{Z}mud${}^{\mathrm{\prime }}$ (1956) showed that the minimum number of irreducible constituents in a faithful complex representation of $G$ coincides with the minimum number of generators of its socle as a normal subgroup; this result can also be extended to arbitrary fields of any characteristic $p$ such that $G$ has no nontrivial normal $p$-subgroup (i.e., over which $G$ admits a faithful completely reducible representation). Rhodes (1969) characterized the finite semigroups admitting a faithful irreducible representation over an arbitrary field as generalized group mapping semigroups over a group admitting a faithful irreducible representation over the field in question. Here, we provide a common generalization of the theorems of \v{Z}mud${}^{\mathrm{\prime }}$ and Rhodes by determining the minimum number of irreducible constituents in a faithful completely reducible representation of a finite semigroup over an arbitrary field (provided that it has one). Our key tool for the semigroup result is a relativized version of \v{Z}mud${}^{\mathrm{\prime }}$'s theorem that determines, given a finite group $G$ and a normal subgroup $N⊲G$, what is the minimum number of irreducible constituents in a completely reducible representation of $G$ whose restriction to $N$ is faithful.