I am a PhD student in the represention theory of finite groups. One of my friends and I solved all exercises in the book [I M Isaacs - Algebra A Graduate Course](https://bookstore.ams.org/view?ProductCode=GSM/100) except for the following exercise in Chapter 9(transfer theory):


**(Exercise 9.4)**


Let $G$ be a finite group and $P\in {\rm Syl}\_p(G)$ 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.


It is equivalent to the following proposition:


**Proposition**


Let $G$ be a finite group and $P\in {\rm Syl}\_p(G)$, then $ {\rm Z}({\rm N}\_G(P))\cap P\cap G'\le P'$.


**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 ${\rm N}\_G(P)$.


In this problem, it is hard to show that if $tz^nt^{-1}\in P$, then $tz^nt^{-1}=z^n$.


I asked a professor in our university, he can't solve it as well.


Here is my idea about this question:


By contradiction, suppose that $z\in G'$, then $z\in {\rm Z}(N)\cap P\cap G'$. 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 results about fusion and focal subgroups are useful.


I checked this question for some classes of finite groups. I guess that it is not true for simple groups.


Could you give us some help? Thank you very much.



I M Isaacs Algebra Exercise 9.4
I think the exercise is incorrect (and so is the hint).


There was a question before about this same exercise at math.stackexchange; the answer there gives a counterexample. Link: [Math.SE](https://math.stackexchange.com/questions/4577477/fusion-in-the-normaliser-of-a-sylow-subgroup/).


