Orbits of irreducible binary forms over GF$(p)$

TL;DR

This paper derives a formula to count the orbits of irreducible binary forms over GF(p) under GL(2,p) action, aiding classification of certain algebraic structures.

Contribution

It provides a new formula for counting orbits of irreducible binary forms over finite fields, with applications in classifying specific algebraic groups.

Findings

Derived a formula for orbit counts of irreducible binary forms.

Applied the formula to classify class two groups of exponent p.

Enhanced understanding of algebraic group structures over finite fields.

Abstract

In this note I give a formula for calculating the number of orbits of irreducible binary forms of degree $n$ over GF$(p)$ under the action of GL$(2,p)$. This formula has applications to the classification of class two groups of exponent $p$ with derived groups of order $p^2$.