主 题: Can the finite simple groups be classified at all?
报告人: Prof.G.Michler (Cornell University)
时 间: 2009-05-15 下午15:00 - 16:00
地 点: 理科一号楼 1114
A finite simple (non abelian) group is called \emph{sporadic} if it is not isomorphic to an alternating group or a finite group of Lie type. In 1981 D. Gorenstein announced the so called classification theorem which asserts that there are exactly 26 sporadic simple groups (up to isomorphism). In 1994 D. Gorenstein, R. Lyons, R. Solomon started a book series published by the Anerican Mathematical Society which aims to provide a complete proof of Gorenstein's assertion. It has been quoted by many authors in several areas of mathematics. But the book series has not been completed up to now. However, R. Solomon's recent survey article ``A brief history of the classification of the finite simple groups" (Bull. Amer. Math. Soc., 2002) contains the following statement: \emph{``Is there a 27th sporadic simple group? I seriously doubt it, but it would be chutzpahdich to assert that a $5000$-page 40-year human endeavor is beyond the possibility of human error"}.
Already in 1979 the late Harvard Professor R. Brauer stated in a survey article that it might be impossible to prove that there are finitely many sporadic simple groups. Their number could even be infinite.
In this lecture I present a new algorithm which constructs simple groups $G$ from indecomposable subgroups $T$ of the general linear groups $GL(n,2)$ over the field $GF(2)$ with 2 elements where $n \geq 2$. In theory, the famous Brauer-Fowler theorem and this algorithm provide a general classification scheme for all finite simple groups $G$ whose Sylow $2$-subgroups $S$ have a non cyclic elementary abelian characteristic subgroup $A$ such that $C_G(A)$ is a $2$-group. By Kondo's work all alternating groups $A_n$ with $n > 4$ satisfy all the conditions of the algorithm. So do many groups of Lie type. In my recent book "Theory of finite simple groups" (2 volumes) $23$ of the known $26$ sporadic simple groups have been constructed in a uniform way by means of the algorithm. For technical reasons the cases of the monster and baby monster have only been outlined. The corresponding $25$ indecomposable subgroups $T$ of the $25$ sporadic groups are contained in $GL(n,2)$ with $3 \le n \le 24$. Since the Mathieu group $M_{11}$ has a semi-dihedral Sylow $2$-subgroup it is covered by the Alperin-Brauer-Gorenstein classification theorem. Thus it can be neglected here.
If the classification theorem were true, then for all other indecomposable subgroups $T$ of all the infinitely many linear groups $GL(m,2)$ a successful application of the algorithm would either construct an alternating group or a finite group of Lie type. By a classical result of D.G. Higman any such group $T$ with a non cyclic Sylow $2$-subgroup has infinitely many indecomposable representations. So it seems to be unpredictable whether the latter general statement can be proved theoretically at all. On the other side, for each given integer $n$ there is an exhaustive search method for all the simple groups $G$ which can be constructed from one of the finitely many indecomposable subgroups $T_{n,k}$ of $GL(n,2)$ by means of the algorithm.
