Result: On finite groups admitting a special noncoprime action: On finite groups admitting a special noncoprime action.

Let \(G\) be a finite solvable group and let \(A\) be a group of automorphisms of \(G\) of prime order \(p\). If the orders of \(A\) and \(G\) are relatively prime, a theorem of the reviewer [\textit{A. Turull}, J. Algebra 86, 555-566 (1984; Zbl 0526.20017)] tells us that the Fitting height of \(G\) is bounded above by the Fitting height of \(C_G(A)\) plus \(2\). The situation is very different when \(|A|\) divides \(|G|\). The present paper shows that one can get a similar result by replacing the coprimeness hypothesis, by a much milder one, but only in the case when \(C_G(A)\) is nilpotent. In particular, the author obtains that the Fitting height of \(G\) is at most \(3\) under the following hypotheses: \(AG\) does not contain any elements of order divisible by \(p^2\) outside of \(G\), and for every element \(\alpha\in AG\), \(\alpha\not\in G\) and \(\alpha\) of order \(p\), then \(C_G(\alpha)\) is nilpotent.