  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$'$ (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$'$ 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$'$'s theorem that determines, given a finite group $G$ and a normal
subgroup $N\lhd G$, what is the minimum number of irreducible constituents in a
completely reducible representation of $G$ whose restriction to $N$ is
faithful.
