On groups admitting a group of automorphisms whose centralizer has bounded rank

We obtain restrictions on the structure of a finite group $G$ with a group of automorphisms $A$ in terms of the order of $A$ and the rank of the fixed-point subgroup $C_G(A)$. When $A$ is regular, that is, $C_G(A)=1$, there are well-known results giving in many cases the solubility of $G$, or bounds for the Fitting height. Some earlier “almost regular” results were deriving the solubility, or bounds for the Fitting height, of a subgroup of index bounded in terms of $|A|$ and $|C_G(A)|$. Now we prove rank analogues of these results: when “almost regular” in the hypothesis is interpreted as a restriction on the rank of $C_G(A)$, it is natural to seek solubility, or nilpotency, or bounds for the Fitting height, of “almost” entire group modulo certain bits of bounded rank. The classification is used to prove almost solubility. For soluble groups the Hall–Higman-type theorems are combined with the theory of powerful $p$-groups to obtain almost nilpotency, or bounds for the Fitting height of a normal subgroup with quotient of bounded rank. Examples are produced showing that some of our results are in a sense best-possible, while certain results on almost regular automorphism have no valid rank analogues. Several open problems are discussed, especially in the case of nilpotent $G$.