Recently, I am studying the book I M Isaacs - Algebra A Graduate Course. I am trying to solve the following exercise:
(Exercise 9.4)
Let $P\in {\rm Syl}_p(G)$ and $N={\rm N}_G(P)$. Suppose $z\in {\rm Z}(N)\cap P$ and $z\notin P'$. Show that $z\notin G'$.
HINT: Use the Transfer evaluation lemma. Note that if $tz^nt^{-1}\in P$, then $tz^nt^{-1}=z^n$ by Lemma 9.12.
Lemma 9.12 (Burnside)
Let $P\in {\rm Syl}_p(G)$ and suppose that $x,y\in {\rm C}_G(P)$ are conjugate in G. Then $x$ and $y$ are conjugate in $N_G(P)$.
In this problem, it is hard to show that if $tz^nt^{-1}\in P$, then $tz^nt^{-1}=z^n$.
Here is my idea about this question:
By contradiction, suppose that $z\in G'$, then $z\in Z(N)\cap P\cap G'$. In order to use Lemma 9.12，We can choose two elements $tz^nt^{-1}$ and $z^n$，So it suffices to show that $tz^nt^{-1}\in{\rm C}_G(P)$. But I don't know whether it is true or not. Maybe some result about fusion is useful.