FINITE -GROUPS ADMITTING -AUTOMORPHISMS WITH FEW FIXED POINTS

Abstract

The following theorem is proved: if a finite -group admits an automorphism of order having exactly fixed points, then it contains a subgroup of -bounded index that is solvable of -bounded derived length. The proof uses Kreknin's theorem stating that a Lie ring admitting a regular (that is, without nontrivial fixed points) automorphism of finite order , is solvable of -bounded derived length . Some techniques from the theory of powerful -groups are also used, especially, from a recent work of Shalev, who proved that, under the hypothesis of the theorem, the derived length of is bounded in terms of , , and . The following general proposition is also used (this proposition is proved on the basis of Kreknin's theorem with the help of the Mal'tsev correspondence, given by the Baker-Hausdorff formula): if a nilpotent group of class admits an automorphism of finite order , then, for some -bounded number , the derived subgroup is contained in the normal closure of the centralizer . The scheme of the proof of the theorem is as follows. Standard arguments show that may be assumed to be a powerful -group. Next, it is proved that is nilpotent of -bounded class. Then the proposition is applied to . There exist explicit upper bounds for the functions from the statement of the theorem. Bibliography: 22 titles.