id: 01557618 dt: j an: 01557618 au: Kondrat’ev, A.S.; Trofimov, V.I. ti: Stabilizers of graph vertices and a strengthened version of the Sims conjecture. so: Dokl. Math. 59, No.1, 113-115 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 364, No.6, 741-743 (1999). py: 1999 pu: Maik Nauka/Interperiodica, Moscow; Pleiades Publishing, New York; Springer, Secaucus, NJ la: EN cc: ut: Thompson-Wielandt theorem; balls; primitive automorphism groups; finite connected graphs; Sims conjecture; conjugate maximal subgroups; normal subgroups ci: li: ab: Let $Γ$ be a non-directed finite connected graph with vertex set $V(Γ)$. For $G\le\AutΓ$, $x\in V(Γ)$ and a non-negative integer $i$, denote by $G_x^{[i]}$ the element stabilizer in $G$ of the closed ball of $Γ$ 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 $V(Γ)$ 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 $i\ge 0$, 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. rv: Victor Mazurov (Novosibirsk)