On elements with restricted eigenvalues in linear groups (Q1906640)

This paper contains three results of different nature. Let \(G\) be a finite linear group over a field of characteristic 0. (1) Suppose that \(G\) contains a \(p\)-element \(g\) with less than \((p-1)/2\) distinct eigenvalues. Then \(g\) belongs to an abelian normal subgroup of \(G\). This proves a conjecture stated by the reviewer [in Algebra, some current trends, Lect. Notes Math. 1352, 206-218 (1988; Zbl 0661.20032)]. (2) Suppose that \(G\) is generated by elements \(X\), \(Y\) such that \(X\) is of odd order, and for every eigenvalue \(e^{i\alpha}\) of \(X\) and \(e^{i\beta}\) of \(Y\) one has \(0\neq\alpha<2\pi/3\), \(0\neq\beta<\pi\). Then \(G\) is a central product \(A\circ B\) where \(A\) is abelian and \(B\) is a central product of copies of \(\text{SL}(2,5)\). (3) Suppose that \(G\) is a primitive irreducible group generated by pseudo-reflections of order \(>2\). The author provides a new proof of an old result of Mitchell saying that \(n\neq 4\).