P-core-free iff there exists a faithful irreducible representation over a field of characteristic p

Statement
The following are equivalent for a finite group $$G$$ and a prime number $$p$$:


 * 1) $$G$$ has no nontrivial normal $$p$$-subgroup.
 * 2) $$G$$ possesses a faithful irreducible representation over a field of characteristic $$p$$.