This is a following-up question of this.
Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups" states:
(i) Let $T$ be a finite $p$-group whose Frattini subgroup is cyclic and central. Then $T'$ has order $1$ or $p$ and there are subgroups $X, Y$ such that $T$ = $X\circ Y$ where $X$ is extraspecial and $Y$ has an abelian maximal subgroup and $\Omega_1(Y)$ is elementary abelian.
(ii) If $T/T'$ is elementary abelian, $Y$ is of the form $p^r$ or $\mathbb{Z}_{p^2} \times p^r$.
The proof for (i):
Notice that $T'$ has order 1 or $p$. The abelian case is trivial, so we assume that $T'$ has order $p$. Let $U \ge T'$ satisfy $U/T'=\Omega_1(T/T')$. Let $E$ be an extraspecial subgroup of $U$ such that $U=Z(U)E$ where $Z(U)$ denotes the center. Then, $[E,T]=T'=Z(E)$ implies that $T=C(E)E$ where $C(E)$ denotes the centralizer. Since $T/U$ is cyclic and $T'$ has order $p$, we see that $|C(E):C(E)\cap C(Z(U))|=1$ or $p$. Since the Frattini subgroup of $T$ is cyclic, the same is true for subgroups and quotients, whence $C(E)\cap C(Z(U))$ is central-by-cyclic, hence abelian. Take $X=E$ and $Y=C(E)$.
Question: I don't really understand the last piece of the argument. Does "being central-by-cyclic" mean being cyclic after quotient out the center? Of which group is the same true for subgroups and quotients? The group $C(E)\cap C(Z(U))$? Any help would be appreciated.
Edit: "central-by-cyclic" has been confirmed. How to go from $|C(E):C(E)\cap C(Z(U))|=1$ or $p$ to $C(E)\cap C(Z(U))$ being central-by-cyclic is still not clear to me..
