A Characterization Of Lie Type Simple²F₄（2⁷）and²F₄（2⁹）

Recently,more and more schoolars are trying to characterize finite simple groups using the properties of finite groups,to our joy,most of them have harvested lots of meaningful results.As we all know,the basement of finite simple groups’ structures is finite simple groups,in order to characterize a finite simple group, studying finite simple groups by the normalizers’orders of their finite groups Sylow subgroups is always always an important method to us.We use the orders of normalizers of the Sylow-p subgroups of the finite simple groups to characterize2F4(27)and2F4(29)which belong to Lie type simple groups.The paper is mainly divided into three chapters.The main content can be listed as follows:Chapter One,introduction,we first introduce some usual symbols,especially the basical concepts and some basical theorems such as group of isomorphism theorem,Sylow theorem,Frattini theorem and so on.Chapter Two,we use N/C theorem by a lot of knowledge involving of elementary number theory,at the same time,computing the orders of the normalizers of p-subgroups.It will give a new characterization of2F4(27)and2F4(29).Thus we reach the following conclusions:Theorem2.1.1:Suppose G is a finite group,and|NG(P)|=|N2F4(27)(R)|,for every primer,where P∈SylrG andR∈Sylr(2F4(27)),then G(?)2F4(27).Theorem2.1.2:suppose G is a FInite group With|G|=|2f4(27)|and|NG(P)|=|N2F4(27)(R)|,there exists a maximum primer,where P∈SylrG andR∈Sylr(2F4(27)),then G(?)(27).Theorem2.2.1:Suppose G is a finite group,and|NG(P)|=|N2F4(29)(R)|,for every primer,where P∈SylrG andR∈Sylr(2F4(29)),thenG(?)2F4(29).Theorem2.2.2:Supp.se G is a finite group with|G|=|2F4(29)|,and|NG％(P)=|N2F4(29)(R)|,there exists a maximum primer,where P∈SylrG and R∈Sylr(2F4(29)),then G(?)2F4(29). Normalizers are found by N/C theorem in this paper.From that point, finite simple groups are characterized.Chapter Three, the direction of further work.