The Existence of Triple Factorizations for Sporadic Groups of Rank 3

Abstract

A finite group G with proper subgroups A and B has triple factorization G = ABA if every element g of G can be represented as g = aba0 , where a and a 0 are from A and b is from B. Such a triple factorization may be sometimes degenerate to AB-factorization. The task of finding triple factorizations for a group is fundamental and can be used for understanding the group structure. For instance, every simple finite group of Lie type has a natural factorization of such a type. Besides, the triple factorization is widely used in the study of graphs, geometries and varieties. The goal of this article is to find triple factorizations for sporadic groups of rank 3. We have proved the existence theorem of ABA-factorization for sporadic simple groups McL and F i22. There exist two rank 3 permutation representations of F i22. We have proved that ABA-factorizations exist in both cases.

Keywords

• group factorization
• sporadic groups
• mclaughlin group
• fisher group