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

This article gives a proof/explanation of the equivalence of multiple definitions for the term p-core-free group
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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 $p$.