Let \(A\) be a finite group acting coprimely on a finite group \(G\). Let \(r\) a fixed prime, suppose that \(A\) is an elementary abelian \(r\)-group and denote by \(A^\#\) the set of non-trivial elements of \(A\). In [Bull. Aust. Math. Soc. 5, No. 2, 281–282 (1971; Zbl 0221.20030)], J. N. Ward demonstrated that if \(|A|\geq r^3\) and \(\mathbf{C}_G(a)\) is nilpotent for every \(a\in A^\#\), then \(G\) is nilpotent, and likewise, a further result due to Ward [loc. cit] shows that if \(|A|=r^2\) and \(G\) is solvable, then \(G\) is metanilpotent. As an analog of Ward’s results [loc. cit], the authors previously proved that if \(|A|\geq r^3\) and \(\mathbf{C}_G(a)\) is \(p\)-nilpotent for each \(a \in A^\#\), then \(G\) is \(p\)-nilpotent. In the paperr under review, the authors address the \(p\)-nilpotent analog of Ward’s result [loc. cit] for the case \(|A|=r^2\). In fact, they extend Ward’s results [loc. cit] to the following more general context.
Let \(\mathfrak{X}\) be a formation consisting of \(p\)-nilpotent groups and let \(p,r\) be two primes. Suppose that an elementary abelian \(r\)-group \(A\) of order \(r^2\) acts coprimely on a \(p\)-solvable group \(G\) in such a way that \(\mathbf{C}_G(a)\) is an \(\mathfrak{X}\)-group for each \(a\in A^\#\). Then \(G^{\mathfrak{X}}\leq \mathbf{O}_{p',p}(G)\), where \(G^\mathfrak{X}\) denotes the \(\mathfrak{X}\)-residual of \(G\).
In fact, the authors assert that the \(p\)-solvability hypothesis in the above theorem can be removed by appealing to a result of R. Guralnick and P. Shumyatsky [Isr. J. Math. 126, 345–362 (2001; Zbl 1050.20013)] which employs the Classification of Finite Simple Groups. Several consequences of the above theorem dealing with the nilpotency class of \(G\), are also obtained in the paper.