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I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that all the permutation modules ($1_H^G$ for $H\leq G$) in characteristic 2 have self-dual socles. Obviously this is true in characteristic 0 where the modules are completely reducible (since $1_H^G$ is self-dual). However, this is not the case in characteristic 3 with $GL(2,3)$ being the smallest example. I also found counterexamples in characteristic 5 (the smallest of order 320), and for non-solvable groups starting with $PSL(3,2)$. However, I didn't find any counterexamples for solvable groups in characteristic 2 (up to $|G|=700$) which I found to be quite odd.

It seems like this should have been conjectured or proven somewhere (if it's true), but I was unable to find any references to it in the literature—likely because I don't know what to search for. So my questions are

  • Is there a counterexample?

  • If not, has this been investigated in the literature?

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The closest work I can think of of this type is a 2010 paper in Journal of Algebra by Natalie Naehrig "On the Endomorphism Rings of Permutation Modules". This was not confined to characteristic 2, nor to solvable groups, and usually considered the permutation module (in characteristic p) on the cosets of a Sylow p-subgroup. However, it is a very rich supply of examples, and a cautionary warning about the rarity of counterexamples (which nevertheless do exist in Naehrig's context).

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