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Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$.

My question is: If $Z(G)$ and $Z(H)$ are isomorphic, then $Z(1+\operatorname{rad}(KG))$ and $Z(1+\operatorname{rad}(KH))$ are ismomorphic and vice versa?

Sven Wirsing
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  • If the centers of G,H are isomorphic, then the mentioned centers do not need to be isomorphic because we can take an arbitrary finite p-group P. As groups we take P and $Z(P)$. There are also other examples of this kind. – Sven Wirsing May 31 '19 at 11:56
  • The other implication is still open. – Sven Wirsing May 31 '19 at 11:57
  • see https://math.stackexchange.com/questions/3244149/two-p-groups-of-exponent-p-with-same-number-of-conjugacy-classes-but-non-isomo/3246592#3246592 for the opposite implication whoch is wrong, too. – Sven Wirsing May 31 '19 at 14:50

1 Answers1

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see both comments; both implications are wrong.

Sven Wirsing
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