# How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?

This is a crosspost from MSE since I haven't found an answer there yet.

I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}_p[G]$-modules for constructing counterexamples. Right now I am just making them in Magma, but I am looking to understand what I'm doing a little better.

Say that I am given a group $G$ which I would like to act on an elementary abelian $p$-group $E$ so that a prescribed proper subgroup $H\leqslant G$ acts fixed point freely on $E$, with $|H|$ and $p$ coprime. So, I construct an absolutely irreducible $\mathbb{F}_p[G]$-module with a trivial fixed point space under restriction to $H$. Often I have other subgroups which I would explicitly like to not act fixed point freely on $E$ (usually faithfully or trivially) in which case I also insist the module have a nontrivial fixed point space under restriction to those subgroups.

I am trying to figure out how to determine the smallest dimension of such a module - in other words, I want to know how big $E$ has to be. Obviously this is going to depend on the structure of $G$, and in particular the structure of $H$. Which group properties determine this, precisely? Does anyone know where I could read about this specifically?

There is some existing theory. For example, one can prove that if $F$ is a Frobenius group with cyclic complement $C$, and $F$ acts faithfully on an elementary abelian $p$-group $E$, then $|C|$ divides the minimum dimension of $E$ over $\mathbb{F}_p$. Are there more general results?

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Hmm. No takers? –  Alexander Gruber Nov 20 '12 at 7:44
Maybe too hard? You could have started with the easier question asking which $H$ act fixed point freely on an elementary abelian $p$-group, or -- if you already know the solution to this question -- given your readers some hints about what you know about $H$. –  Someone Nov 20 '12 at 13:15
I don't know the answer. I guess it must be too hard. –  Alexander Gruber Nov 20 '12 at 15:21
Do you know anything nontrivial about finite $p'$-groups $H$ acting fixed point freely on elementary abelian $p$-groups? –  Someone Nov 20 '12 at 15:49
Well if the entire $p'$-group $H$ acts fixed point freely on an elementary abelian group $E$ then $E\rtimes H$ is a Frobenius group with kernel $E$. Evidently Frobenius groups with abelian kernels have been classified up to isomorphism in Ch. 13 of this memoir: google.com/… but it is way above my head I am not sure whether one can get the minimum dimension from it. –  Alexander Gruber Nov 21 '12 at 6:18