On a conjecture of Thompson (Q5905463)

Let \(A\) be a finite \(p\)-group of order \(p^ n\) acting on a finite \(p\)- soluble group, and suppose it acts fixed-point-freely on every \(A\)- invariant \(p'\)-section. The conjecture in question is that in this situation, the \(p\)-length \(l_ p(G)\) of \(G\) is bounded by a linear function of \(n\). It is known that if \(p\) is odd, then \(l_ p(G)\leq 2^{2+n}-1\) [\textit{A. Rae} and the reviewer, Bull. Lond. Math. Soc. 5, 197-198 (1973; Zbl 0273.20015)]. The author has conjectured that the correct bound is \(n+1\), and proved this if \(A\) is cyclic of odd order [Math. Z. 196, 323-329 (1987; Zbl 0612.20008)]. In this paper this is established for \(p\geq 5\), provided \(G\) satisfies certain quite strong restrictions. Theorem. With the hypotheses stated in the first sentence, assume that \(p\geq 5\), and that \(G\) is of \(p\)-splitting type with respect to \(G\). If 2 divides \(| G|\) and \(p\) is a Mersenne prime, assume further that \(A\) is \(C_ p\wr C_ p\)-free. Then \(l_ p(G)\leq n+1\). This bound is best possible. One of the difficulties that beset attempts to prove theorems of this type is the shortage of useful \(A\)-invariant subgroups of \(G\). Another is the absence of an ``invariance of centralizers'' theorem for the action of \(A\) on sections of \(G\). The latter is to some extent compensated for by the hypothesis on the action of \(A\) on the \(A\)-invariant \(p'\)- sections. The \(p\)-splitting type hypothesis ensures a reasonable supply of \(A\)-invariant subgroups. Even with it, as often is the case in theorems of this type, the proof is quite involved, and finally comes down to character-theoretic questions, regular orbit theorems, and the like, which are too technical to discuss in detail here. We give the definition of \(p\)-splitting type. Definition. Let the finite group \(A\) act on the finite group \(G\). Then \(G\) is of \(p\)-splitting type with respect to \(G\), if (a) \(G=P_ 1P_ 2\dots P_ n\), where \(P_{i+1}\) is \(AP_ 1\dots P_ i\)-invariant for all \(i\). (b) \(P_ i\) is either a \(p\)-group or a \(p'\)-group and \((| P_ i|,| P_{i+1}|)=1\). (c) \(P_ i\) acts faithfully on \(P_{i+1}\). (d) \(AP_ 1\dots P_ i\) normalizes a Sylow \(r\)-subgroup of \(P_{i+1}\) for each prime \(r\).