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I was reading Professor Yukov Berkovich' paper "On Subgroups of Finite $p$-groups" when I stumled upon the following theorem:

Let $G$ be a nonabelian $p$-group with cyclic Frattini subgroup, $|\Phi(G)|>p$ and let $\Phi_0$ be the subgroup of order $p$ in $\Phi(G)$. Let $Z$ be cyclic subgroup of maximal order in $G$ containing $\Phi(G)$; then $|Z|=p|\Phi(G)|$. Set $\Delta_1:=$ {$H<G|[H:Z]=p$}. Suppose that every $H\in \Delta_1$ contains a normal abelian subgroup of type $(p,p)$. Then $\Phi(G)\leq Z(G)$ and $G=(A_1***A_s)Z(G)$, where $A_i$ are minimal nonabelian and $\Phi_0=G'=A_i'$ for all $i$. If $|A_i|>p^3$, then $A_i$ has a cyclic subgroup of index $p$. Moreover, $G=AZ$, where $A$ is generated by all normal subgroups of $G$ of type $(p,p)$ containing $\Phi_0$.

Here $"*"$ denotes the central product. In his paper, Yukov does not give a reference, and when I contacted him by mail he was still unable to do so. I think the theorem can play a vital role in the bit of research I am doing in character theory, so I would very much like to see the proof.

Can somebody give me a reference for the above theorem (or perhaps a similar statement)?

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    $\begingroup$ Isn't it Theorem 6 from Berkovich' paper (freely available on sciencedirect.com/science/article/pii/S0021869399980043) which is proved on page 215? $\endgroup$ – tj_ Feb 15 at 12:14
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    $\begingroup$ Added: According to the old title (changed by YCor for whatever reason), you are looking for a reference of the theorem. Why can't you just reference Berkovich if no earlier reference is at hand? $\endgroup$ – tj_ Feb 15 at 12:19

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