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

It is proved that if a finite p-soluble group G admits an automorphism ' of order pn having at most m fixed points on every '-invariant elementary abelian p0-section of G, then the p-length of G is bounded above in terms of pn and m; if in addition the group G is soluble, then the Fitting height of G is bounded above in terms of pn and m. It is also proved that if a finite soluble group G admits an automorphism of order paqb for some primes p; q, then the Fitting height of G is bounded above in terms of j j and jCG( )j