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 $φ$ 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 $p^aq^b$ for some primes $p,q$, then the Fitting height of $G$ is bounded above in terms of $|ψ|$ and $|C_G(ψ)|$.

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