Return to Revisions

One possible argument is to compare this with $T/C(Z(U))$ as follows. We have $$|C(E):C(E)\cap C(Z(U))|=|C(E)C(Z(U)):C(Z(U))|$$ $$\leqslant|T:C(Z(U))|\leqslant|T:U\Phi(T)|$$ (since $\Phi(T)$ is central in $T$, it centralises $Z(U)$). Now $T/U$ is cyclic and $T/\Phi(T)$ has exponent $p$, so $T/U\Phi(T)$ is cyclic of exponent dividing $p$, so has order $1$ or $p$.