Group rings isomorphic over F_p, but not over Z_p ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:28:45Z http://mathoverflow.net/feeds/question/104941 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104941/group-rings-isomorphic-over-f-p-but-not-over-z-p Group rings isomorphic over F_p, but not over Z_p ? Matthias Künzer 2012-08-17T19:51:25Z 2012-08-17T19:51:25Z <p>Suppose given a prime $p$.</p> <p>Question: Do there exist finite groups $G$ and $H$ such that ${\bf F}_p G$ is isomorphic to ${\bf F}_p H$, but such that ${\bf Z}_p G$ is not isomorphic to ${\bf Z}_p H$ ?</p> <p>Variants: Suppose given $s\geqslant 2$ and replace ${\bf Z}_p$ resp. ${\bf F}_p$ by ${\bf Z}/p^s$.</p> <p>Variant: Suppose $G$ and $H$ to be $p$-groups. (It is unknown whether there are nonisomorphic $p$-groups with isomorphic group rings over ${\bf F}_p$ , but still, maybe someone knows an argument in favour of ${\bf F}_p G \simeq {\bf F}_p H$ $\Rightarrow$ ${\bf Z}_p G \simeq {\bf Z}_p H$ in this case?)</p>