# Curves with more than one inner Galois point

## TL;DR

This paper classifies irreducible plane curves with two distinct simple inner Galois points, detailing the structure of their Galois groups and providing explicit examples for each classified group.

## Contribution

It offers a complete classification of the Galois groups for curves with two simple inner Galois points, extending understanding of their algebraic and geometric properties.

## Key findings

- Classification of Galois groups for such curves
- Explicit examples for each Galois group type
- Deeper group-theoretic analysis applied

## Abstract

Let $\mathcal{C}$ be an irreducible plane curve of $\text{PG}(2,\mathbb{K})$ where $\mathbb{K}$ is an algebraically closed field of characteristic $p\geq 0$. A point $Q\in \mathcal{C}$ is an inner Galois point for $\mathcal{C}$ if the projection $\pi_Q$ from $Q$ is Galois. Assume that $\mathcal{C}$ has two different inner Galois points $Q_1$ and $Q_2$, both simple. Let $G_1$ and $G_2$ be the respective Galois groups. Under the assumption that $G_i$ fixes $Q_i$, for $i=1,2$, we provide a complete classification of $G=\langle G_1,G_2 \rangle$ and we exhibit a curve for each such $G$. Our proof relies on deeper results from group theory.