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Gasch\"utz (1954) proved that a finite group $G$ has a faithful irreduciblecomplex representation if and only if its socle is generated by a singleelement as a normal subgroup; this result extends to arbitrary fields ofcharacteristic $p$ so long as $G$ has no nontrivial normal $p$-subgroup.\v{Z}mud$'$ (1956) showed that the minimum number of irreducible constituentsin a faithful complex representation of $G$ coincides with the minimum numberof generators of its socle as a normal subgroup; this result can also beextended to arbitrary fields of any characteristic $p$ such that $G$ has nonontrivial normal $p$-subgroup (i.e., over which $G$ admits a faithfulcompletely reducible representation).  Rhodes (1969) characterized the finite semigroups admitting a faithfulirreducible representation over an arbitrary field as generalized group mappingsemigroups over a group admitting a faithful irreducible representation overthe field in question. Here, we provide a common generalization of the theoremsof \v{Z}mud$'$ and Rhodes by determining the minimum number of irreducibleconstituents in a faithful completely reducible representation of a finitesemigroup over an arbitrary field (provided that it has one).  Our key tool for the semigroup result is a relativized version of\v{Z}mud$'$'s theorem that determines, given a finite group $G$ and a normalsubgroup $N\lhd G$, what is the minimum number of irreducible constituents in acompletely reducible representation of $G$ whose restriction to $N$ isfaithful.

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