I have the following situation. A subgroup $U\leq G$ of a finite group $G$ and its $O^p$, $\textit{i.e.}$ the minimal normal subgroup for which the quotient $U/O^p(U)$ is a $p$-group. Then I consider the $p$-Sylow of the quotient group $N_G(O^p(U))/O^p(U)$, which is of the form $U_p/O^p(U)$, for some subgroup $U_p$ placed as follows:
$$O^p(U)\leq U\leq U_p\leq N_G(O^p(U)).$$
Here the inclusion $U\leq N_G(O^p(U))$ is given by the property of normalizer, while $U\leq U_p$ by maximality of the $p$-Sylow subgroup. Now, certainly the centralizer $C_G(U)$ contains $C_G(U_p)$, and there's no guarantee for the converse inclusion. However, I'd like if something could be said if we allow cojugation by elements of the normalizer. Is that true that for every $x\in C_G(U)$ there's a $g\in N_G(O^p(U))$ such that $x\in gC_G(U_p)g^{-1}$?
Thanks in advance
Best Answer
Let's look for the simplest type of example that fits the description, so we'll take $G = U_p$ to be a nonabelian $p$-group, and $U=Z(G)$ (so $O^p(U)=1$).
Then $C_G(U_p) = U$ is normal and hence $gC_G(U_p)g^{-1} = C_G(U_p)=U$ for all $g \in G$. So any $x \in U_p \setminus U$ gives a counterexample.