Finite soluble and nilpotent groups with a restriction on the rank of the centralizer of an automorphism of prime order (Q1586971)

In 1976, P.~Fong proved (mod CFSG) that a finite group containing an element of prime order \(p\) with centralizer of order \(m\) has the soluble radical of \((p,m)\)-bounded index. In 1981, B.~Hartley and T.~Meixner proved that the index of the Fitting subgroup of a soluble group with the above conditions is \((p,m)\)-bounded as well. In 1985, the author proved that a nilpotent group with the above conditions contains a subgroup of \(p\)-bounded nilpotency class and \((p,m)\)-bounded index. So, if a finite group \(G\) contains an element of prime order \(p\) with centralizer of order \(m\) then \(G\) contains a nilpotent subgroup of \(p\)-bounded nilpotency class and \((p,m)\)-bounded index. The author deals with the following problem: Can the order bounds be substituted by bounds of ranks? Some new approaches for soluble and nilpotent groups are obtained.