On groups satisfying a positive law in fixed points.
The goal of the paper under review is to prove the following result. Let \(q\) be a prime. Suppose the elementary Abelian \(q\)-group \(A\) of order \(q^2\) acts on the finite \(q'\)-group \(G\) so that for any non-identity \(a,b\in A\) the subgroup \(\langle C_G(a),C_G(b)\rangle\) satisfies a law which is positive (i.e., it does not include inverses), of degree \(n\). Then \(G\) satisfies a positive law of degree bounded by a function of \(q\) and \(n\) only. This result is part of a large body of literature which studies the influence of the structure of the centralizer of a coprime action on the structure of a finite group. Celebrated results in this area include \textit{J.~G.~Thompson}'s proof that a finite group \(G\) admitting a fixed-point-free automorphism of prime order \(q\) is nilpotent [Proc. Natl. Acad. Sci. USA 45, 578-581 (1959; Zbl 0086.25101)], and G.~Higman's bound on the nilpotence class of such a \(G\) in terms of \(q\) alone.
scientific article; zbMATH DE number 5602651