Order three normalizers of 2-groups

ABSTRACT

This paper examines order three elements of finite groups which normalize no nontrivial 2-subgroup. The motivation for finding such elements arises out of a problem in modular representation theory. The question of when these elements appear in the almost simple groups was posed by Robinson in the context of studying 2-blocks of defect zero. For the almost simple groups, a complete classification of order three elements with this property is determined. On the basis of this result, necessary conditions are then given for the existence of such elements in a large class of finite groups.

KEYWORDS:

• Blocks of defect zero
• centralizers of order three elements
• finite simple groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 20E32
• Secondary 20G99

Acknowledgments

The author would like to thank Geoff Robinson for suggesting the question considered in this paper, and Robert Guralnick and Gunter Malle for helpful comments and suggestions on an earlier version of this manuscript. This research was partially funded by NSF grant DMS-1265297.