#### Event Title

### The Isomorphy Classes of the Special Orthogonal Group of Low Dimension in Characteristic 2

#### Session Number

Project ID: MATH 02

#### Advisor(s)

Dr. Ellen Ziliak; Benedictine University

#### Discipline

Mathematics

#### Start Date

19-4-2023 11:25 AM

#### End Date

19-4-2023 11:50 AM

#### Abstract

In 1926, Cartan introduced the concept of real symmetric spaces as a class of homogeneous Riemannian manifolds, where a symmetric space is defined as the homogeneous space $G/H$ with $G$ a reductive group and $H=G^{\theta}=\{g\in G|\theta(g)=g\}$, the fixed-point group of an $n$-automorphism. Of special interest are automorphism of order 2, also called involutions. In this project we focused on classifying the involutions for the $SO(3,\mathbb{F}_{2^k})$ and $SO(4,\mathbb{F}_{2^k})$. Our results should provide a base case for the general theory for n-dimensional matrices.

*Note: The above math text can be compiled using LaTeX.*

The Isomorphy Classes of the Special Orthogonal Group of Low Dimension in Characteristic 2

In 1926, Cartan introduced the concept of real symmetric spaces as a class of homogeneous Riemannian manifolds, where a symmetric space is defined as the homogeneous space $G/H$ with $G$ a reductive group and $H=G^{\theta}=\{g\in G|\theta(g)=g\}$, the fixed-point group of an $n$-automorphism. Of special interest are automorphism of order 2, also called involutions. In this project we focused on classifying the involutions for the $SO(3,\mathbb{F}_{2^k})$ and $SO(4,\mathbb{F}_{2^k})$. Our results should provide a base case for the general theory for n-dimensional matrices.

*Note: The above math text can be compiled using LaTeX.*