Orders of inner-diagonal automorphisms of some simple groups of Lie type

Abstract

Abstract We refer to the set of element orders of a finite group 𝐺 as the spectrum of 𝐺. For the simple groups PSL n ⁢ ( q ) \mathrm{PSL}_{n}(q) , PSU n ⁢ ( q ) \mathrm{PSU}_{n}(q) , E 6 ⁢ ( q ) E_{6}(q) , and E 6 2 ⁢ ( q ) {}^{2}E_{6}(q) , we describe the spectra of extensions of these groups by diagonal automorphisms.