Summary

Group theory

Definition

A group is a set $G$ equipped with a binary operation $\ast$ that satisfies the following axioms:

• Closure: For all $a,b\in G$, $a\ast b\in G$.
• Associativity: For all $a,b,c\in G$, $(a\ast b)\ast c=a\ast (b\ast c)$.
• Identity element: There exists an element $e\in G$ such that for all $a\in G$, $a\ast e=e\ast a=a$.
• Inverse element: For all $a\in G$, there exists an element $a^{-1}\in G$ such that $a\ast a^{-1}=a^{-1}\ast a=e$.

Introduction

Group theory is the study of group, which is a fundamental algebraic structure. Groups are used to study symmetry, geometry. This is used in define an abstract elements of symmetry, which is a group of transformations that preserve the structure of an object.

Abelian groups

An Abelian group, also known as a commutative group, is a group in which the binary operation is commutative, i.e., $a\ast b=b\ast a$ for all $a,b\in G$. Abelian groups play a crucial role in algebra and have many interesting properties.

Definition

An abelian group is a group $G$ in which the binary operation is commutative, i.e., $a\ast b=b\ast a$ for all $a,b\in G$.

Exmaples

Examples of groups

Example

• The set of integers $\mathbb{Z}$ under addition is a group.
• The set of non-zero rational numbers ${\mathbb{Q}}^{\ast }$ under multiplication is a group. (except 0 as it does not have an inverse)
• The set of $n\times n$ invertible matrices under matrix multiplication is a group.
• The set of symmetries of a square is a group with operators being rotation and reflection.

Example of proof

Example

Prove that the set of integers $\mathbb{Z}$ under addition is a group.

• Closure: For all $a,b\in \mathbb{Z}$, $a+b\in \mathbb{Z}$. (Sum of two integers is an integer)
• Associativity: For all $a,b,c\in \mathbb{Z}$, $(a+b)+c=a+(b+c)$. (Addition is associative)
• Identity element: The identity element is 0, as $a+0=0+a=a$ for all $a\in \mathbb{Z}$.
• Inverse element: For all $a\in \mathbb{Z}$, the inverse element is $-a$, as $a+(-a)=(-a)+a=0$.