Grün's first theorem on the focal subgroup

Statement
Suppose $$G$$ is a finite group and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Let $$P_0$$ be the focal subgroup of $$P$$ in $$G$$. Then:

$$P_0 = \langle P \cap N_G(P)', P \cap Q' \mid Q \in \operatorname{Syl}_p(G) \rangle$$.

In other words, $$P_0$$ is generated by the intersection between $$P$$ and the commutator subgroup of its normalizer, along with the intersection between $$P$$ and the commutator subgroups of all $$p$$-Sylow subgroups.

Facts used

 * 1) uses::Focal subgroup theorem
 * 2) uses::Alperin's fusion theorem in terms of tame intersections

Proof
Given: A finite group $$G$$ with $$p$$-Sylow subgroup $$P$$ having focal subgroup $$P_0$$.

To prove: $$P_0 = P_1$$ where:

$$P_1 = \langle P \cap N_G(P)', P \cap Q' \mid Q \in \operatorname{Syl}_p(G) \rangle$$.

Proof:

Proof that $$P_1 \le P_0$$

 * 1) $$P_1 \le P$$: This is clear, since all the subgroups used to generate $$P_1$$ are contained in $$P$$.
 * 2) $$P_1 \le [G,G]$$: All the subgroups used to generate $$P_1$$ are contained in the commutator subgroup of some subgroup, which in turn is contained in $$[G,G]$$. Thus, $$P_1 \le [G,G]$$.
 * 3) $$P_1 \le P_0$$: By the previous two steps, $$P_1 \le P \cap [G,G]$$. By the focal subgroup theorem (fact (1)), $$P_0 = P \cap [G,G]$$, so $$P_1 \le P_0$$.