On the rank and exponent of the fixed points of coprime actions (Q2116376)

Let \(A\) be a finite group acting coprimely on a \(p\)-group \(P\). The author investigates cases when \(A\) acting trivially on certain abelian subgroups of \(P\) implies that \(A\) acts trivially on \(P\). The author has obtained a character theory free proof [Arch. Math. 113, No. 3, 225--227 (2019; Zbl 07089432)] of a result in this area by \textit{I. M. Isaacs} and \textit{G. Navarro} [Arch. Math. 95, No. 3, 207--211 (2010; Zbl 1206.20022)]. In this paper, the author refines these results, especially in the case \(p = 2\). The paper also explores some related issues involving the rank, the special rank, and the exponent of the groups involved.