Commutator Calculus for Wreath Product Groups

Abstract

We introduce a class of function spaces consisting of integer-valued functions of several integers which coordinatize the elements of certain subgroups of some finite regular wreath product groups. On each function space, we define operators which correspond to forming certain commutators relevent to computing the upper central series. We define an automorphism of the function space which enables us to define a class of subgroups that is useful for describing the upper central series of certain finite regular wreath product p-groups. Our results describe fundamental, interesting, and useful relationships between the automorphism and the operators. We describe some applications.

Key Words:

• Binomial coefficient
• Commutator
• Upper central series
• Wreath product

2010 Mathematics Subject Classification:

• 20F12
• 20E22
• 20D15

ACKNOWLEDGMENTS

We thank the referee for helpful suggestions for the proofs of Lemmas 3.8 and 5.1.

Notes

Communicated by P. Tiep.