The well-known Margulis-Platonov conjecture describes the normal subgroup structure of an algebraic group \(G\) over a number field \(K\). The conjecture has been established for almost all isotropic groups and most anisotropic groups except for those of type \(A_n\). Inner forms of anisotropic groups \(G\) of type \(A_n\) have the form \(\text{SL}_{1,D}\), the reduced norm 1 group of a finite dimensional division algebra \(D\) over \(K\) [see V. P. Platonov and A. S. Rapinchuk, Algebraic groups and number theory, Pure Appl. Math. 139, Academic Press (1994; Zbl 0841.20046)].
In this case A. Potapchik and A. Rapinchuk [Proc. Indian Acad. Sci., Math. Sci. 106, No. 4, 329-368 (1996; Zbl 0879.20027)] showed that if \(\text{SL}_{1,D}\) fails to satisfy the conjecture then there exists a proper normal subgroup \(N\) of the multiplicative group \(D^*\) of \(D\) such that \(D^*/N\) is a nonabelian finite simple group.
The commuting graph \(\Delta(H)\) of a finite group \(H\) is the graph whose vertex set is \(H\setminus Z(H)\) and whose edges are pairs \(\{h,g\}\subseteq H\setminus Z(H)\), such that \(h\not=g\) and \([h,g]\in Z(H)\); the diameter of \(\Delta(H)\) is denoted by \(\text{diam}(\Delta(H))\). If \(d\) is the distance function on \(\Delta(H)\), then \(\Delta(H)\) is balanced if there exists \(x,y\in\Delta(H)\) such that the distances \(d(x,y)\), \(d(x,xy)\), \(d(y,xy)\), \(d(x,x^{-1}y)\), \(d(y,x^{-1}y)\) are all larger than 3.
The first author [Ann. Math. (2) 149, No. 1, 219-251 (1999; Zbl 0935.20009)] showed that if \(L\) is a nonabelian finite simple group and either \(\text{diam}(\Delta(H))>4\), or \(\Delta(H)\) is balanced, then \(L\) cannot be isomorphic to a quotient of \(D^*\). Consequently, the Margulis-Platonov conjecture for inner forms of anisotropic groups of type \(A_n\) is resolved by the following theorem, which is the main result of this paper: If \(L\) is a nonabelian finite simple group then either \(\text{diam}(\Delta(L))>4\), or \(\Delta(L)\) is balanced.
To prove the theorem the authors establish results on the commuting graph of a nonabelian finite simple group \(L\) (these results may have independent interest). In particular, they prove that if \(L\) is classical, then \(\Delta(L)\) is balanced and \(\text{diam}(\Delta(L))\geq 4\); if \(L\not\cong E_7(q)\) is either an exceptional group of Lie type, or a sporadic group, then \(\Delta(L)\) is disconnected; if \(L=E_7(q)\), then \(\Delta(L)\) is balanced; if \(L=A_n\), \(n\geq 5\), then \(\text{diam}(\Delta(L))>4\).
The authors note that the commuting graph of \(L\) is connected if and only if the prime graph of \(L\) is connected. Thus the nonabelian finite simple groups \(L\) for which \(\Delta(L)\) is disconnected are known by the papers of J. S. Williams [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)], the reviewer [Mat. Sb. 180, No. 6, 787-797 (1989; Zbl 0691.20013)] and N. Iiyori and H. Yamaki [J. Algebra 155, No. 2, 335-343 (1993; Zbl 0799.20016); Corrections ibid. 181, No. 2, 659 (1996)].