Enumerating classes and characters of p-groups

TL;DR

This paper develops formulas to count conjugacy classes and irreducible characters in finite p-groups of nilpotency class less than p, unifying and extending previous results using advanced algebraic tools.

Contribution

It introduces general formulas for enumeration in p-groups of certain classes, generalizing prior results with the Lazard correspondence and Kirillov orbit method.

Findings

Unified enumeration formulas for conjugacy classes and characters

Extended results to generalized relatively free p-groups

Applicable to p-groups of nilpotency class less than p

Abstract

We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite p-groups of nilpotency class less than p. This allows us to unify and generalize a number of existing enumerative results, and to obtain new such results for generalizations of relatively free p-groups of exponent p. Our main tools are the Lazard correspondence and the Kirillov orbit method.