Simultaneous Conjugacy Classes of Finite $p$-groups of rank $\leq 5$

TL;DR

This paper studies the counting of simultaneous conjugacy classes in finite p-groups of rank ≤ 5, analyzing their generating functions and normalized forms to understand their algebraic structure.

Contribution

It introduces the analysis of normalized generating functions for conjugacy classes in finite p-groups of small rank, extending understanding of their algebraic properties.

Findings

Generating functions are rational functions of t.

Normalized functions reveal structural properties of p-groups.

Results apply specifically to groups of rank ≤ 5.

Abstract

For a finite group $G$, we consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $\alpha_{G,n}$ denote the number of simultaneous conjugacy classes of $n$-tuples, and $\beta_{G,n}$ the number of simultaneous conjugacy classes of commuting $n$-tuples in $G$. The generating functions $A_G(t) = \sum_{n\geq 0} \alpha_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0} \beta_{G,n}t^n$ are rational functions of $t$. This paper concern studied of normalized functions $A_G(t/|G|)$ and $B_G(t/|G|)$ for finite $p$-groups of rank at most $5$.