Further remarks on linear groups (Q1188268)

The author investigates restrictions on the eigenvalues of elements in finite subgroups of \(GL(n,\mathbb{C})\) which ensure that these elements commute with all their conjugates. In particular, the following corollary holds: Let \(G\) be a finite subgroup of \(GL(n,\mathbb{C})\) with no composition factor isomorphic to \(A_ 5\), and suppose that \(x\) is an element of \(G\) whose eigenvalues all lie on an arc of length less than \(6\pi/13\) on \(S^ 1\). Then \(x\) commutes with all its \(G\)-conjugates.