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)?
