Unrecognizability by spectrum of finite simple orthogonal groups of dimension nine

摘要

The spectrum of a finite group is the set of its elementsorders. A group G is said to be unrecognizable by spectrum if there areinfinitely many pairwise non-isomorphic finite groups having the samespectrum as G. We prove that the simple orthogonal group O9(q) hasthe same spectrum as V oO??8 (q) where V is the natural 8-dimensionalmodule of the simple orthogonal group O??8 (q), and in particular O9(q)is unrecognizable by spectrum. Note that for q = 2, the result wasproved earlier by Mazurov and Moghaddamfar.