# Representation of finite group and Classifying finite simple group

My question is quite clear as the title of it. I had studied theory of representation of finite group and I also had known something about the program of classifying finite simple group, and in general, the problems of two subjects is not far different, but in fact, I have not seen any mathematical paragraph that tells us about the connection(of these two subject).

But, I think the two subject both give us the structure of finite group. Could any MSer point it out for me?

Thanks.

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Will this link help, ams.org/notices/199502/solomon.pdf ? But, there's very little about the connection, but points out to the work of many people in this direction –  user21436 Dec 21 '11 at 16:23
Your title does not at all make it clear what you want to ask... –  Mariano Suárez-Alvarez Dec 21 '11 at 20:02
Sorrry, Mariano. I in my head thought about the title "the bridge between representation theory of finite group and classifying finite simple group, but I did wrote down the other name without awaring. –  Knumber10 Dec 22 '11 at 2:00

Representation theory of finite groups over the complex numbers yields information about groups which can be be determined by group characters. The most important characters in this context are the irreducible characters, and complete information about these can be encoded within the character table, which is a $k \times k$ array, where $k$ is the number of conjugacy classes of the group in question, say $G.$ It can be determined by inspection from the character table whether there are any normal subgroups of $G$ other than the trivial group and $G$ itself, ie whether $G$ is simple or not. However, it is sometimes possible to use remarkably small amounts of character-theoretic information to prove the non-simplicity of $G.$ There is a long tradition of using the embedding of certain subgroups of $G$ to produce enough character-theoretic information to show that $G$ is not simple. Two early examples of such theorems were Burnside's $p^a q^b$-theorem and Frobenius theorem about subgroups of $G$ which have trivial intersection with all their distinct conjugates. This tradition continued until the 1960s and beyond, and was give fresh impetus by the work of Richard Brauer, who showed how to use modular methods to further refine character-theoretic information. These methods were particularly effective when dealing with simple groups of low $2$-rank, that is without elementary Abelian $2$-subgroups of order $8$ or more. In particular, character theory played a prominent role in the proof by Feit and Thompson of the solvability of finite groups of odd order, and in the classification of finite simple groups with dihedral or semi-dihedral Sylow $2$-subgroups (as well as proving that no finite simple group has generalized quaternion Sylow $2$-subgroups). Another high point of the use of character theory was Glauberman's $Z^{*}$-theorem, which proved that if $G$ is a finite non-Abelian simple group, and $t$ is an involution of $G,$ then $t$ commutes with one of its other conjugates, which was used as a starting point for many group-theoretic analyses in the classification programme. It is probably fair to say that in the later stages of the classification programme, ordinary character theory played less of a role, with the honorable exception of the $Z^{*}$-theorem. On the other hand, most of the infinite families of finite simple groups are best described in terms of natural representations over finite fields.