# Simultaneous Conjugacy Classes as Combinatorial Invariants of Finite Groups

## TL;DR

This paper investigates the counting of simultaneous conjugacy classes in finite groups, establishing their generating functions as rational, and explores their relation to group structure and isoclinism classes.

## Contribution

It introduces generating functions for counting conjugacy classes, proves their rationality, and links these invariants to the group's class equation and isoclinism.

## Key findings

- Generating functions are rational functions of t.
- Growth rate of conjugacy classes is exponential with base |G|.
- Normalized invariants are preserved under isoclinism.

## Abstract

Let $G$ be a finite group. 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 $t$. We show that $A_G(t)$ determines and is completely determined by the class equation of $G$. We show that $\alpha_{G,n}$ grows exponentially with growth factor equal to the cardinality of $G$, whereas $\beta_{G,n}$ grows exponentially with growth factor equal to the maximum cardinality of an abelian subgroup of $G$. The functions $A_G(t)$ and $B_G(t)$ may be regarded as combinatorial invariants of the finite group $G$. We study dependencies amongst these invariants and the notion of isoclinism for finite groups. We prove that the normalized functions $A_G(t/|G|)$ and $B_G(t/|G|)$ are invariants of isoclinism families.