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**On finite homomorphic images of the multiplicative group of a division algebra.**
*(English)*
Zbl 0935.20009

The commuting graph \(\Delta(X)\) of a finite group \(X\) is the graph whose vertex set is \(X\setminus\{1\}\) and whose edges are the pairs \(\{a,b\}\) of commuting elements \(a,b\in X\) with \(a\neq b\) and \(a,b\neq 1\). This graph is called balanced if there exist vertices \(x\), \(y\) such that the distances \(d(x,y)\), \(d(x,x^{\pm 1}y)\), \(d(y,x^{\pm 1}y)\) are all bigger than \(3\).

The main result of the paper is that the multiplicative group of a finite dimensional division algebra over an arbitrary field does not have as homomorphic image any nonabelian finite simple group \(X\) with the property that \(\Delta(X)\) is balanced or that the diameter of \(\Delta(X)\) is strictly larger than \(4\). On the other hand, the author announces that the commuting graph of any nonabelian finite simple group is either balanced or has diameter strictly larger than \(4\).

A proof (using the classification of finite simple groups) is in [Y. Segev, G. M. Seitz, “Anisotropic groups of type \(A_n\) and the commuting graph of finite simple groups.” (Preprint)]. Together, these results prove a conjecture of Potapchik and Rapinchuk: No nonabelian finite simple group is a homomorphic image of the multiplicative group of a finite dimensional division algebra. As a consequence, using the work of A. Potapchik and A. Rapinchuk [Proc. Indian Acad. Sci., Math. Sci. 106, No. 4, 329-368 (1996; Zbl 0879.20027)], a conjecture of Margulis and Platonov on the normal subgroup structure of the group of rational points of simple, simply connected algebraic groups over algebraic number fields is established in the case of anisotropic groups of inner type \(A_n\).

The main result of the paper is that the multiplicative group of a finite dimensional division algebra over an arbitrary field does not have as homomorphic image any nonabelian finite simple group \(X\) with the property that \(\Delta(X)\) is balanced or that the diameter of \(\Delta(X)\) is strictly larger than \(4\). On the other hand, the author announces that the commuting graph of any nonabelian finite simple group is either balanced or has diameter strictly larger than \(4\).

A proof (using the classification of finite simple groups) is in [Y. Segev, G. M. Seitz, “Anisotropic groups of type \(A_n\) and the commuting graph of finite simple groups.” (Preprint)]. Together, these results prove a conjecture of Potapchik and Rapinchuk: No nonabelian finite simple group is a homomorphic image of the multiplicative group of a finite dimensional division algebra. As a consequence, using the work of A. Potapchik and A. Rapinchuk [Proc. Indian Acad. Sci., Math. Sci. 106, No. 4, 329-368 (1996; Zbl 0879.20027)], a conjecture of Margulis and Platonov on the normal subgroup structure of the group of rational points of simple, simply connected algebraic groups over algebraic number fields is established in the case of anisotropic groups of inner type \(A_n\).

Reviewer: J.-P.Tignol (Louvain-La-Neuve)

### MSC:

20D05 | Finite simple groups and their classification |

16K20 | Finite-dimensional division rings |

16U60 | Units, groups of units (associative rings and algebras) |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

20E32 | Simple groups |

20G30 | Linear algebraic groups over global fields and their integers |

20E36 | Automorphisms of infinite groups |