A Characterization Of F₄(q), Q=2,3,4

In recent years, using subgroups and quotients to describe the nature of finite groups has become a hot topic. Many scholars have made a lot of work, and got a large number of results.Based on a lot of professional literature and the relevant papers, and under the guidance of the instructor, I used N / C theorem to determine the order of the normalizer subgroup, then used it to characterize simple group F4 (q ), q =2,3,4 which is the innovations of this article.This paper is divided into three chapters, as follows:Chapter One describes the development of using the orders of normalizers of Sylow subgroups to characterize the finite simple groups, the prime graph and related forms of finite simple groups,.Finally, we propose research questions.Chapter Two proves the thorem : G≌F₄ (q), q =2,3,4 if and only if |N_G(R)|=|N_G(S)|, R∈Syl_pG, S∈Syl_p F₄(q ), p is prime, using basic finite group theorem and elementary number theorem to determine the order of Sylow subgroup's normalizer.Chapter Three refers to the further study.