Minimum polynomials of the elements of prime order in representations of quasi-simple groups.
Let \(p\) be a prime number, \(G\leq\text{GL}(n,\mathbb{C})\). We call \(G\) an \(N_p\)-group if it contains a matrix \(g\) such that \(g^p\) is a scalar matrix, \(g\) has at most \(p-1\) distinct eigenvalues and \(g\) does not belong to a proper normal subgroup of \(G\). As no \(N_2\)-group exists for \(n>1\) we assume \(p>2\). In this paper quasi-simple groups of \(N_p\)-type are determined. The cases where \(g\) has at most \(p-2\) distinct eigenvalues or a Sylow \(p\)-subgroup of \(G\) is cyclic are considered by the author earlier (Theorems 2.1 and 2.2). The main result considers the case where \(g\) has exactly \(p-1\) distinct eigenvalues and a Sylow \(p\)-subgroup of \(G\) is noncyclic (Theorem 1.1).
scientific article; zbMATH DE number 5365177
scientific article