Finite Groups Which Factor as Product of an Alternating Group and a Symmetric Group

Abstract

We will find the structure of finite groups G = AB where A and B are subgroups of G with A ≅ A _r , the alternating group on r letters, and B ≅ S _n , the symmetric group on n letters, where r, n ≥ 5.

Key Words:

• Factorization
• Simple group
• Symmetric group

2000 AMS Subject Classification:

• 20D40

Acknowledgments

This research was carried out at Yale University while the author was on a sabbatical leave from the University of Tehran in the year 2001. The author would like to thank the research council of the University of Tehran for supporting this research. He also would like to express his deep gratitude to Professor Walter Feit for many useful discussions he had with him.

Notes

^#Communicated by A. Yu Olshamskii.