P-core-free group

Definition
Let $$G$$ be a finite group and $$p$$ be a prime number. We say that $$G$$ is $$p$$-core-free if it satisfies the following equivalent conditions:


 * The $$p$$-core (i.e., the largest normal $$p$$-subgroup of $$G$$) is trivial. The $$p$$-core is also termed the Sylow-core, and is the normal core of any Sylow subgroup. It also equals the pi-core where $$\pi = \{ p \}$$.
 * $$G$$ has no nontrivial normal $$p$$-subgroup.
 * $$G$$ possesses a faithful irreducible representation over a field of characteristic $$p$$.

Equivalence of definitions
The first two definitions are clearly equivalent. For equivalence with the third definition, refer p-core-free iff there exists a faithful irreducible representation over a field of characteristic p.