Group

Given a Binary Operation $\ast$ : $G\times G\to G$ on a set $G$, the pair $(G,\ast )$ is defined as a group if it satisfies the following conditions:

1. $\ast$ is associative.
2. There exists an identity element $e\in G$.
3. For each element $a\in G$, there exists an inverse element $a^{-1}\in G$.

Example

• $G{L}_2(\mathbb{R})$ is the set of all invertible 2×2 matrices, and with matrix multiplication, it forms a group.

Note

• Commutativity: For $\forall a,b\in G$, $a\ast b=b\ast a$
• Associativity: For $\forall a,b,c\in G$, $(a\ast b)\ast c=a\ast (b\ast c)$
• Identity: $\exists e\in G$ s.t. $\forall ainG$, $a\ast e=e\ast a=a$
• Inverse: For $\forall a\in G$, $\exists a^{-1}\in G$ s.t. $a\ast a^{-1}=a^{-1}\ast a=e$

Abelian Group

A group $G$ under $\ast$ is an abelian group if $\ast$ is commutative.

Example

• $(\mathbb{R},+)$ is an abelian group.
• $({\mathbb{R}}^{\ast },\cdot )$ is an abelian group. (${\mathbb{R}}^{\ast }=\mathbb{R}$ \ $0$)
• $(\mathbb{R},\cdot )$ is NOT a group. Because there is no inverse to 0.
• $(\mathbb{N},+)$ is NOT a group. Because there is no identity and inverse.