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Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random matrices from it, takes their characteristic polynomials, and hands them to you. Is there a criterion that will let you figure out if $H=G$?

Existing theorems I've found use the existence of transvections, which can't (as far as I can tell) be deduced from the above. Or they work over a field of characteristic $0$, where we have more Galois groups to work with. Some also work over the algebraic closure, which is a weaker hypothesis and I'm hoping it strengthens this result. This has applications to large image theorems and experimental mathematics.

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  • $\begingroup$ I guess you would like to impose some upper bound on the number of random matrices to be picked? $\endgroup$ – Stefan Kohl Apr 28 '17 at 22:12
  • $\begingroup$ It should look at a finite number of characteristic polynomials. $\endgroup$ – Watson Ladd Apr 28 '17 at 23:51
  • $\begingroup$ Can you, please, clarify 2 points: 1) Do you literally mean random matrices (i.i.d. and) picked according to some probability distribution? 2) What exactly would constitute a criterion in this context? $\endgroup$ – Victor Protsak Apr 29 '17 at 1:38
  • $\begingroup$ Here is an example of a criterion I would like: if you have five matrices whose characteristic polynomials have galois group of order n, and which don't share roots, they generate the whole group. Maybe I need to reword the question. What I don't want: if $H$ contains a transvection, or if $H$ contains a matrix conjugate to a given matrix, as these aren't looking at the characteristic polynomials. $\endgroup$ – Watson Ladd Apr 29 '17 at 14:12

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