Nikolaus Conference 2025

It is well known that for a group \(G\) and a subgroup \(A\) of \(G\) it is impossible to cover \(G\) with the conjugates of \(A\). Thus, instead of the conjugates of \(A\) we take the conjugates of the coset \(Ax\) in \(G\) and check if the union of \((Ax)^g\) covers \(G-\{1\}\) for \(g \in G\). Moreover, if \((Ax)^g\) covers \(G\) for all \(Ax\) in Cos\((G:A)\), we say that \((G,A)\) is CCI. We are aiming to classify all such pairs. It has been proven by Baumeister-Kaplan-Levy that this can be reduced down to the case where \(A\) is maximal in \(G\) and so that the action of \(G\) on Cos\((G:A)\) is primitive. And they showed that \((G,A)\) is CCI if \(G\) is 2-transitive. By O'Nan-Scott Theorem and CFSG, we see that \(G\) is either an affine group or almost simple. In the paper of Baumeister-Kaplan-Levy, it has been shown that affine CCI groups are 2-transitive. Thus, it remains to consider the almost simple groups. By employing the knowledge of buildings, representation theory and Aschbacher-Dynkin theorem, we prove that, apart from finitely many small cases, the CCI almost simple groups are 2-transitive.