I was reading the proof of the following theorem but I cannot understand how to use the class equation as he wants me to.
Theorem Suppose that $G=HK$ where $H$ is a normal locally finite $p'$-subgroup of $G$ and $K$ is a finite non-cyclic abelian $p$-group. Then $$H=\langle C_H(x)|1\neq x\in K\rangle$$
Proof Let $h\in H$. Then $F=\langle h^K\rangle$ is finite and is a subgroup of $H$ since $H$ is normal in $G$. Let $q\neq p$ be a prime, by Sylow theorem the number of Sylow $q$-subgroups of $F$ is prime to $q$. By the class equation and orbit stabilizer theorem $K\leq N_G(P)$ for some $P\in Syl_qF$.
Why is it this? I think that $|G:N_G(P)|=|Syl_q(F)|$ should be the part "orbit stabilizer theorem", but what about the "class equation" then? How can one use it?
I don't know, but you could prove that $K \le N_G(P)$ for some $P \in {\rm Syl}_q(F)$ as follows.
Let $P \in {\rm Syl}_q(F)$. Then by the Frattini Argument, $FK = FN_{FK}(P)$. So $N_{FK}(P)$ must contain some Sylow $p$-subgroup $K'$ of $FK$. Then $K = (K')^g$ for some $g \in FK$ and $K \le N_G(P^g)$.