Fischer groups based on vertex operator algebras

研究了24次最大Fischer群的自然作用的顶点算子代数VF，发现了VF的c=6/7 Virasoro顶点算子子代数与Fischer群的转置之间的对应关系.基于这种对应关系，我们得到了一个框架，它使我们能够解释Fischer群的3转置性质以及最大Fischer群与扩展的E6 Dynkin图之间的神秘关系。我们还研究了顶点算子超代数VB，它提供了Baby-monster零星单群的自然作用，并发现了Baby-monster和VB的c=7/10 Virasoro顶点算子子代数的转置之间的一一对应.通过我们的结果还解释了Baby-monster的4-转置性质及其与扩展E7 Dynkin图的关系。我们导出了推广的Griess代数的迹公式，推广了Matsu-Norton迹公式，并发现它在偶发有限单群上的应用是卓有成效的.

We study the vertex operator algebra VF which affords a natural action of the largest Fischer group of degree 24.We discovered a correspondence between the transpositions of the Fischer group and c=6/7 Virasoro vertex operator subalgebras of VF. Based on this correspondence, we obtained a framework which enables us to explain the 3 transposition property of the Fischer group and a mysterious relation between the largest Fischer group and the extended E6 Dynkin diagram. We also study the vertex operator superalgebra VB which affords a natural action of the Baby-monster sporadic simple group and found a one-to-one correspondence between the transpositions of the Baby-monster and c=7/10 Virasoro vertex operator subalgebras of VB. The 4-transposition property of the Baby-monster and its relation to the extended E7 Dynkin diagram is also explained via our results. We derived trace formulae for the extended Griess algebras which generalize the Matsu-Norton trace formulae and we found its fruitful applications to the sporadic finite simple groups.