On element-centralizers in finite groups. (Q1047897)

Let \(G\) be a finite group. Denote by \(\text{Cent}(G)\) the set of centralizers \(C_G(x)\) of elements \(x\) of \(G\). The author of the paper under review determines \(|\text{Cent}(G)|\) for all finite simple groups in which every proper subgroup is solvable. Using this result, he proves that a finite non-Abelian simple group \(G\) with \(|\text{Cent}(G)|\leq 73\) is isomorphic to the alternating group \(A_5\) of degree \(5\).