Diving into Group Theory: Foundations for Equivariant Neural Networks
![Diving into Group Theory: Foundations for Equivariant Neural Networks](/images/content/group-theory-foundations-equivariant-neural-networks/group_theory.png)
What is a Group?
A group is an algebraic structure consisting of a set of elements and a binary operation that combines any two elements to form a third element. Let's say we have a set of elements and a binary operation "", then we can say that is a group if the following properties are satisfied:
- Closure: The closure property states that for any two elements , the result of the binary operation is also an element of . In other words, the binary operation "" is closed under the set . Mathematically, we can write this as:
- Associativity: The binary operation "" is associative if for any three elements , the following holds true:
- Identity: There exists an element such that for any element , the following holds true:
- Inverse: For any element , there exists an unique element such that the following holds true:
A group is said to be abelian if the binary operation "" is commutative. In other words, for any two elements , the following holds true:
Now, as we have a basic understanding of what a group is, let's look at some examples of groups.
Examples of Groups
-
Integers under addition: The set of integers under addition is a group. The closure property is satisfied as the sum of any two integers is also an integer. The associativity property is also satisfied as the addition of integers is associative. The identity element is as for any integer . The inverse of an integer is as .
-
Integers modulo under addition : The set of integers modulo under addition is a group. The closure property is satisfied as the sum of any two integers modulo is also an integer modulo . The associativity property is also satisfied as the addition of integers modulo is associative. The identity element is as for any integer modulo . The inverse of an integer modulo is as .
-
Vectors under addition : Every vector space is a group under addition. The closure property is satisfied as the sum of any two vectors is also a vector. The associativity property is also satisfied as the addition of vectors is associative. The identity element is the zero vector as for any vector . The inverse of a vector is as .
Order of a Group
The order of a group is the number of elements in the group. The order of a group is denoted by . For example, the order of the group is .
Order of an Element
The order of an element of a group is the smallest positive integer such that , where is the identity element of the group . The order of an element is denoted by . For example, the order of the element in the group is . Confused? Let's break it down.
Let's say we have the set {0, 1, 2, \dots, n-1} and we want to find the order of the element 1. We know that the identity element of the group is 0. So, we need to find the smallest positive integer such that . Let's try to find the order of the element 1 for .
![Mir Sazzat Hossain](/_next/image?url=%2Fimages%2Fuser.png&w=3840&q=75)
About Mir Sazzat Hossain
Mir Sazzat Hossain is a Research Assistant at the Independent University of Bangladesh's Center for Computation and Data Science (CCDS).
Comments
Do you have a problem, want to share feedback, or discuss further ideas? Feel free to leave a comment here! Please stick to English. This comment thread directly maps to a discussion on GitHub, so you can also comment there if you prefer.
Instead of authenticating the giscus application, you can also comment directly on GitHub.
Related Articles
The Poisson Distribution: Your Key to Predicting the Unforeseeable
The blog post explores the Poisson distribution, a statistical distribution commonly used in various fields, including astronomy, to model random events. It explains the properties of the distribution, its real-world applications, and provides a step-by-step guide on how to visualize the Poisson distribution using Python. The post also discusses the issue of Poisson noise in astronomical observations and presents a practical example of how to calculate and deal with it using Python.
Dockerize Your Data Science Workflow: A Step-by-Step Guide to Setting Up Jupyter Lab on Your Private Linux Machine
Learn how to set up and deploy a Dockerized Jupyter Lab on a private Linux server, providing a secure environment for data science work. This step-by-step guide covers the installation and configuration of Docker, creation of a Jupyter Lab container, and tips for managing conda environments and accessing the Jupyter Lab interface.