docs/group.md

content/Basic Algebra/Basic Algebra.md

@@ -24,4 +24,4 @@
 - [[Discrete logarithm]]
 - [[ECDLP]]
 - [[Ideal]]
-- [[Abelian Group]]
+

content/Basic Algebra/Group.md

@@ -6,6 +6,28 @@ Given a [[Binary Operation]] $*$ : $G \times G \rightarrow G$ on a set $G$, the
 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
+### Example
 
-- $GL_{2}(\mathbb{R})$ is the set of all invertible 2×2 matrices, and with matrix multiplication, it forms a group.
+- $GL_{2}(\mathbb{R})$ is the set of all invertible 2×2 matrices, and with matrix multiplication, it forms a group.
+
+<br/>
+
+**Note**
+
+- Commutativity: For $ \forall a, b \in G$, $a * b = b * a$
+- Associativity: For $ \forall a, b, c \in G$, $(a * b) * c = a * (b * c)$
+- Identity: $\exists  e \in G$ s.t. $\forall a in G$, $a * e = e * a = a$
+- Inverse: For $\forall a \in G$, $\exists  a^{-1} \in G$ s.t. $a * a^{-1} = a^{-1} * a = e$
+
+<br/>
+
+# Abelian Group
+
+A group $G$ under $*$ is an abelian group if $*$ is commutative.
+
+### Example
+
+- $(\mathbb{R}, +)$ is an abelian group.
+- $(\mathbb{R}^*, \cdot )$ is an abelian group. ($\mathbb{R}^* = \mathbb{R}$ \ ${0}$)
+- $(\mathbb{R}, \cdot )$ is **NOT** a group. Because there is no inverse to 0.
+- $(\mathbb{N}, +)$ is **NOT** a group. Because there is no identity and inverse.