Hi everyone!


Given a finite simple group $S$, is it always possible to find two elements $x,y \in S$ with the property that for every $a,b \in S$ we have $\langle x^a,y^b \rangle = S$?


More in general, given a finite almost simple group $X$ with socle $S$, is it possible to find two elements $x,y \in X$ with the property that for every $a,b \in X$ we have $\langle x^a,y^b \rangle \supseteq S$?


I can deal with the alternating case, but I don't know where to look for results about the general case.


Thank you for any contribution.



Conjugation stable generating sets in almost simple groups
This property is known as "being invariably generated by $x$ and $y$" in the literature (though it has not been so extensively studied). For a nonabelian finite simple group, Kantor, Lubotzky and Shalev have shown that, indeed, such $x$ and $y$ exist. (This is Theorem 1.3 in <https://arxiv.org/abs/1010.5722> , which contains a lot of interesting material on similar questions).


