This is a follow up to this question

Let $G_{1}$ and $G_{2}$ be two finite UCS p-groups with the following conditions:

1- $\vert G_{1}\vert=\vert G_{2}\vert=p^{2n}$;

2- $\Phi(G_{1})\cong\Phi(G_{2})\cong\underbrace{\mathbb{Z}_{p}\times\mathbb{Z}_{p}\times\dots\times\mathbb{Z}_{p}}_{n\,\,times}$;

3- $G_{1}$ and $G_{2}$ have the same number of minimal subgroups.

Can we prove that $G_{1}\cong G_{2}$? Is there any counterexamples?

  • $\begingroup$ I added some conditions to the problem. However if somebody has any result on the first question, I am eagerly waiting to see it, since my main question is the first one. $\endgroup$ – H.Shahsavari Feb 1 at 18:22
  • 1
    $\begingroup$ Counterexamples for $p=2$ and $n=2k+1\ge5$ are provided in: Graham Higman, Suzuki $2$-groups, Illinois J. Math. 7 (1963), 79--96 $\endgroup$ – Richard Lyons Feb 3 at 21:40
  • $\begingroup$ @ Professor Lyons. Many thanks for your valuable comment. $\endgroup$ – H.Shahsavari Feb 8 at 17:38

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