1. finite simple groups We know that there is a basic concept-group in the development of mathematics. Group is also the most basic concept in mathematics. How to study the structure of groups? The simplest method is to discuss its subgroups, and then gradually construct larger groups from the structure of small groups. The most important group in a group is a finite group, which is a very difficult subject and needs special methods and concepts to study. Let G be a group and g∈G be a subgroup. For any g∈G, gH- 1g ∈H, h is called nomal. The existence of normal subgroups can turn the study of G into the study of subgroup H and quotient group G/H, so naturally there is a question, what is a finite simple group? A simple group has no other nontrivial normal subgroups except itself and unit element. Mathematically, it is called a simple group, but it is not simple at all. A profound theorem of finite group theory is Fei-Thompson theorem: the order (number) of a noncommutative simple group (that is, the number of elements in the group) is even. Even more unusual, except for some big categories (prime cyclic group Zp, staggered group An (n >; =5), Lie type simple group), and then 26 fragmentary finite simple groups (scattered in simple groups and discrete simple groups) are found. Now we know that the order of the largest dispersed simple group is

4 1 20 9 6 2 3 54 2 3 5 7 1 1 13 17 19 23 29 3 1 4 1 47 59 7 1 =808,0 17..= 1054

This is a large simple group, which was discovered by two mathematicians, B.Fisher and R.L.Griess. Mathematicians call it a monster. D.Gorenstein, the authoritative mathematician of simple groups, thinks that finite simple groups are all here, which is of course a good result in mathematics. Just as chemists determine all the elements and physicists determine all the structures of nucleons, a single group is determined. But there is a disadvantage here. Gorenstein didn't write a certificate. He said that if the proof is written, it will be at least 1000 pages, and anyway, the proof of 1000 pages is prone to mistakes. But Gorenstein added that it doesn't matter. If there is a mistake, it can be remedied. Do you believe it or not? Some people in mathematics doubt whether such proof is necessary. With the advent of computers, many problems can be proved in large numbers, and whether it is necessary to prove them strictly has become a controversial issue in mathematics. This argument seems insoluble at present. Mr Duan Xuefu is an old friend of mine and an expert in finite group theory. Maybe we can ask his opinion. Personally, I find this question difficult to answer. But mathematicians have a freedom. When you can't do a problem or don't like to do it, you don't have to invest at all. You just need to do something you can or like.