# Normal subgroups of $p$-groups

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 $$\Delta_1:=$$ {$$H}. Suppose that every $$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)?

• Isn't it Theorem 6 from Berkovich' paper (freely available on sciencedirect.com/science/article/pii/S0021869399980043) which is proved on page 215? – tj_ Feb 15 at 12:14
• 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? – tj_ Feb 15 at 12:19