On finite soluble groups with almost fixed-point-free automorphisms of noncoprime order

Abstract

Abstract. It is proved that if a finite p-soluble group G admits an automorphism ϕ of order p n having at most m fixed points on every ϕ-invariant elementary abelian p ′ -section of G, then the p-length of G is bounded above in terms of p n and m; if in addition the group G is soluble, then the Fitting height of G is bounded above in terms of p n and m. It is also proved that if a finite soluble group G admits an automorphism ψ of order p a q b for some primes p, q, then the Fitting height of G is bounded above in terms of |ψ| and |C G (ψ)|.