On eigenvalues of elements of finite linear groups
Let \(ph(p)\) be the maximal real number with the property that any \(p\)- element of a finite subgroup of \(\text{GL}(n,\mathbb{C})\) whose eigenvalues all lie on an arc of length less than \(ph(p)\) on the unit circle commutes with all its conjugates (this number was introduced by \textit{G. R. Robinson} [J. Algebra 149, 32-45 (1992; Zbl 0796.20036)]). The author invokes the classification of finite simple groups to determine \(ph(p)\) for every prime \(p\) greater than 5. Namely, let \(p\) be an odd prime. Let \(n^ +\) be the length of the shortest interval on the real line which contains a representative for all the quadratic residues (mod \(p\)). Let \(n^ -\) be the length of the shortest interval on the real line which contains a representative for all the quadratic non- residues (mod \(p\)). The following statement is proved: Theorem. \(ph(p) = \min \{n^ +,n^ -\} \cdot 2\pi/p\).
finite subgroup of \(\text{GL}(n,\mathbb{C})\)
scientific article; zbMATH DE number 589929