1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.