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Is there a prime $p$ and an infinite simple $p$-group $G$ such that for any field $K$ of characteristic $p$ the only irreducible $KG$-module, whether finite or infinite dimensional, is trivial (that is, the augmentation ideal and Jacobson radical of $KG$ coincide)?

If this has appeared in print, a reference would be appreciated.

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    $\begingroup$ According to math.wisc.edu/~passman/trento.pdf such a group cannot be finitely generated. $\endgroup$ Mar 3, 2015 at 18:18
  • $\begingroup$ @BenjaminSteinberg I think the same ideas show that for such a group the Jacobson radical (a.k.a. augmentation ideal) must be an idempotent ideal of $KG$. $\endgroup$ Mar 5, 2015 at 12:03

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