On the Number of Distinct Functional Graphs of Affine-Linear Transformations over Finite Fields

TL;DR

This paper investigates the count of non-isomorphic functional graphs generated by affine-linear transformations over finite fields, providing bounds for large dimensions and implications for conjugacy classes in symmetric groups.

Contribution

It establishes new upper and lower bounds on the number of such graphs and relates these bounds to conjugacy class counts in symmetric groups.

Findings

Derived bounds for the number of non-isomorphic functional graphs

Connected bounds to conjugacy classes in symmetric groups

Enhanced understanding of affine-linear transformation structures

Abstract

We study the number of non-isomorphic functional graphs of affine-linear transformations from (\F_q)^n to itself, and we prove upper and lower bounds on this quantity for n large. As a corollary to our result, we prove bounds on the number of conjugacy classes in the symmetric group S_{q^n} that intersect AGL_n(q).