Let $G$ be finite group and suppose that $P\in Sylp_p(G)$ and that $g\in P$ has order $p$. If $g\in G',$ but $g\notin P'$, Show that $g^t\in P'$ for some element $t\in G$ with $t\notin P$.($5.B1$)
Let $\overline V$ be the transfer map from $G$ to $P/P'$. As $g\in G'$, $ \overline {V(g)}=\bar 1$.
By the transfer evaluation lemma, $$V(g)=\prod_{t\in T_0} g^t \in P'.$$
Thus, the product of some conjugates of $g$ lies in $P'$.
But I do not see that it causes $g^t\in P'$ for some $t$.
Any help would be appreciated.