Nilpotent periodic groups with an almost regular automorphism of prime order (Q1114008)

It is proved that there exist two natural-valued functions f(p,n), g(p,n) for p prime and n natural arguments such that every nilpotent periodic group G having an automorphism of prime order p with precisely n fixed points contains a subgroup H such that the nilpotency class of H is not greater than f(p,n) and \(| G:H| \leq g(p,n)\) (Theorem 1). As a corollary of this result, a locally soluble periodic group G containing an element of prime order with \(| C_ G(a)| <\infty\) is almost nilpotent. Theorem 1 is deduced from the following proposition. Every soluble periodic group G of solvability class s having an automorphism of prime order p with precisely n fixed points contains a nilpotent subgroup H such that the nilpotency class of H and the index \(| G:H|\) are bounded by functions depending of p, n, s (Theorem 2).