Simultaneous Conjugacy Classes as Combinatorial Invariants of Finite Groups

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 $A_G(t)={\sum }_{n\ge 0}{\alpha }_{G,n}{t}^n,$ and $B_G(t)={\sum }_{n\ge 0}{\beta }_{G,n}{t}^n$ 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.

arXiv comment11 pages