Characterizations Of Lie Type Simple Groups E₈（13） And E₈（17）

Finite simple groups constitute the basis of finite groups.With the number of properties of finite simple groups to characterize a finite simple group,Also it is always an important method to study finite groups by the normalizers’orders of their Sylow subgroups.We use the orders of normalizers of the Sylow-p subgroups of the finite simple groups to characterize E8(13)and E8(17) which belong to Lie type slmpe groups.The paper is divided into three chapters.The main content can be listed as follows.Chapter One, introduction, we first introduce some usual symbols,the basic concepts and some basic theorems such as isomorphic theorem, Sylow theorem,Frattini theorem and so on.Chapter Two, using N/C theorem and some knowledge of elementary number theory, we compute the order of the normalizer of p-subgroups, it gives two new Characterizations of E8(13)andE8(17).Thus we reach the following conclusions:Theorem1Suppose G is a finite group with|G|=|E8(13)|and|NG(P)|=|NE8(13)(R)|for the largest prime r of|G|,where P∈SylrG andR∈Sylr,(E8(13)), then G(?)E8(13).Theorem2Suppose G is a finite group with|NG(P)|=|NE8(13)(R)|for every prime r,where P∈SylrG andR∈Sylr(E8(13)),then G(?)E8(13).Theorem3Suppose G is a finite group with|G|=|E8(17)|and|NG(P)|=|NE8(17)(R)|for the largest prime r of|G|,where P∈SylrG andR∈Sylr(E8(17)), then G(?)E8(17).Theorem4suppose G is a finite group with|NG(P)|=|NE8(17)(R)|for every prime r,where P∈SylrG andR∈Sylr(E8(17)),then G(?)E8(17).Normalizers are found by N/C theorem in this paper.From that point,finite simple groups are characterized.Chapter Three,The intend of further work.