Characterizations Of Lie Type Simple Groups E₈（7）and E₈（11）

Suppose G is a finite group,for E8(7)and E8(11),this article try to use different ways to characterize a finite simple group with the number of properties of finite simple groups,Maily using the impact of the results and the nature ofE8(7)and E8(11) by the order of their sylow subgroups’normalizers,thus get two new Characterizations of E8(7)and E8(11).Theorem.1Suppose G is a finite group with|G|=|E8(7)|and|NG(P)|=|NE8(7)(R)|,for the largest prime r of|G|,where P∈SylrG, andR∈Sylr(E8(7)), then G(?)E8(7).Theorem.2Suppose G is a finite group with|NG(P)|=|NE8(7)(R)|for every prime r,where P∈Sylr,G andR∈Sylr,(E8(7)),then G(?)E8(7).Theorem.3Suppose G is a finite group with|G|=|E8(11)|and|NG(P)|=|NE8(11)(R)|,for The largest primer of|G|,where P∈Sylr,G andR∈Sylr(E8(11)), then G三E8(11).Theorem.4Suppose G is a finite group with|NG.(P)|=|NE8(11)(R)|for every primer,where P∈Sylr,G andR∈Sylr,(E8(11)),then G(?)E8(11).Normalizers are found by N/C theorem in this paper.From that point,finite simple groups are characterized.