Normal \(p\)-complements and fixed elements.
It is well-known that if \(K\) is a group of odd order acting on a 2-group \(P\) such that \(K\) fixes all elements of order 2 and 4 in \(P\), then \(K\) acts trivially on \(P\). Recently \textit{G. Navarro} and \textit{L. Sanus} [J. Algebra 320, No. 6, 2451-2454 (2008; Zbl 1159.20009)] proved, using character theory, that \(K\) acts trivially on \(P\) if it fixes all rational elements of \(P\). Noting that elements of order 2 are rational and elements of order 4 are rational if and only if they are real, the authors prove, again using character theory, the following ``intersection'' of these results: \(K\) acts trivially on \(P\) if it fixes all elements of order 2 and all real elements of order 4 in \(P\).
scientific article; zbMATH DE number 5797318