The authors consider a finite group \(G\) with a group of automorphisms \(A\) that is almost regular in the sense of rank, that is, the fixed point subgroup \(C_G(A)\) has given rank \(r\). Recall that a finite group has rank \(r\) if all its subgroups can be generated by \(r\) elements. By a famous theorem of Thompson, if a group \(G\) admits a regular group of automorphisms \(A\) (that is, \(C_G(A)=1\)) and \(A\) is of prime order \(p\), then \(G\) is nilpotent; moreover, by Higman’s theorem the nilpotency class of \(G\) is bounded in terms of \(p\) only. It follows from the Classification that a finite group admitting a regular group of automorphisms of coprime order is soluble.
Most of the known generalizations of the above theorems are about groups \(G\) with an almost regular group of automorphisms \(A\) in the sense of order. This means that the fixed point subgroup \(C_G(A)\) has given order \(n\) and the consequences are sought as restrictions on the structure of the group \(G\) in terms of \(|A|\) and \(n\). For example, P. Fong [Osaka J. Math. 13, 483-489 (1976; Zbl 0372.20010)] proved that if \(A\) is of prime order \(p\) and \(|C_G(A)|=n\), then the quotient \(G/S(G)\) by the soluble radical has order bounded in terms of \(p\) and \(n\). When both \(G\) and \(A\) are soluble of coprime orders, A. Turull [J. Algebra 86, 555-566 (1984; Zbl 0526.20017)] and B. Hartley and I. M. Isaacs [J. Algebra 131, No. 1, 342-358 (1990; Zbl 0703.20023)] proved that \(G\) has a normal subgroup \(H\) of index bounded in terms of \(|A|\) and \(|C_G(A)|\) such that the Fitting height of \(H\) is bounded in terms of \(|A|\).
In the present paper the authors obtain rank analogues of these results. They prove that if \((|A|,|G|)=1\), then there is a bound for the rank of the quotient \(G/S(G)\) by the soluble radical in terms of \(|A|\) and the rank of \(C_G(A)\). They also prove a similar result for the orders: if \((|A|,|G|)=1\), then there is a bound for \(|G/S(G)|\) in terms of \(|C_G(A)|\) and \(|A|\). When both \(G\) and \(A\) are soluble, they show that there is a normal subgroup \(H\) such that the quotient \(G/H\) has rank bounded in terms of the rank of \(C_G(A)\) and \(|A|\) and the Fitting height of \(H\) is bounded in terms of \(|A|\). The last result extends an earlier theorem of the authors, where the case of \(A\) of prime order was considered [see E. I. Khukhro and V. D. Mazurov, J. Algebra 301, No. 2, 474-492 (2006; Zbl 1108.20019)].
The Classification is used to prove almost solubility. The proof of the theorem on soluble groups combines the Hall-Higman type theorems with the theory of powerful \(p\)-groups.
For the entire collection see [Zbl 1105.20300].