Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

Abstract

Abstract A subgroup H of a group G is said to be pronormal in G if H and H g {H^{g}} are conjugate in 〈 H , H g 〉 {\langle H,H^{g}\rangle} for every g ∈ G {g\in G} . In this paper, we determine the finite simple groups of type E 6 ⁢ ( q ) {E_{6}(q)} and E 6 2 ⁢ ( q ) {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.