Exact Factorizations of Sporadic Simple Groups

Abstract

A group G is said to have an exact factorization if there exist proper subgroups A_i for $i=1,2,\dots ,n$ such that $G=A_1{A}_2\dots A_n$ and $\left|G\right|=\left|A_1\right|\left|A_2\right|\dots \left|A_n\right|$. The number n is called length of this factorization. An exact factorization of length 3 is called exact triple factorization. In this article, we show the existence of exact factorizations of seven sporadic simple groups ${\mathrm{J}}_1,{\mathrm{J}}_2,\text{HS},\text{McL},\text{He},\text{Ru}$ and ${\text{Co}}_3$. Our factorizations for five groups are exact triple. There are no reported factorizations for the groups ${\mathrm{J}}_1,\text{McL}$ and ${\text{Co}}_3$. We will present an exact triple factorization for ${\text{Co}}_3$ and exact factorizations for ${\mathrm{J}}_1$ and McL of length four.

Keywords:

• factorization
• sporadic simple group; permutation representation

2010 AMS SUBJECT CLASSIFICATION:

• 20D08
• 20D40

Acknowledgment

I would like to thank the anonymous referee who read the paper carefully and proposed corrections.