Stabilizers of graph vertices and a strengthened version of the Sims conjecture (Q1594297)

Let \(\Gamma\) be a non-directed finite connected graph with vertex set \(V(\Gamma)\). For \(G\leq\Aut\Gamma\), \(x\in V(\Gamma)\) and a non-negative integer \(i\), denote by \(G_x^{[i]}\) the element stabilizer in \(G\) of the closed ball of \(\Gamma\) of radius \(i\) (in the natural metric) centered at \(x\). The authors give a scheme of a proof (using the classification of the finite simple groups) that if \(G\) is primitive on \(V(\Gamma)\) then \(G_x^{[6]}=1\) (Theorem 1). This result, which is a strengthened version of the well-known Sims conjecture, is a consequence of the following Theorem 2. Let \(G\) be a finite group and \(M_1=M_1^{(0)}\), \(M_2=M_2^{(0)}\) conjugate maximal subgroups of \(G\). For \(i\geq 0\), define recursively \(M_1^{(i+1)}=\bigcap_{x\in M_1}(M_1^{(i)}\cap M_2^{(i)})^x\) and \(M_2^{(i+1)}=\bigcap_{x\in M_2}(M_1^{(i)}\cap M_2^{(i)})^x\). Then \(M_1^{(6)}\) coincides with \(M_2^{(6)}\) and is a normal subgroup of \(G\). Examples show that the constant 6 in Theorems 1 and 2 can not be reduced.