Quasiprimitve linear groups with quadratic elements (Q5958877)

A semisimple complex linear transformation with exactly two distinct eigenvalues is called a quadratic element. The author determines the finite irreducible linear groups generated by quadratic elements of order 3 for which the multiplicities of the eigenvalues are distinct. He also presents examples of groups having a quasiprimitive irreducible representation of order 3 for which the multiplicities of the distinct eigenvalues are different. Among them are the Weil modules for \(\text{Sp}(2n,3)\) and \(\text{U}(n,2)\).NEWLINENEWLINENEWLINEThese results are used to obtain bounds on the degree of an irreducible primitive linear group containing such elements. More exactly, if \(G\) is an irreducible primitive or quasiprimitive linear group containing a quadratic element of drop \(r\) and order at least \(3\), then the degree of \(G\) is at most \(4r\). If the element is of order distinct from \(3\), the degree is \(2r\).NEWLINENEWLINENEWLINEMoreover, there are no quasiprimitive or primitive irreducible linear groups containing quadratic elements of order 4 with eigenvalues \(\{1,\pm i\}\) with distinct multiplicities where \(i\) is a primitive fourth root of 1.