From 036de99da94f91e69f5c131f25ee5f07212bfef3 Mon Sep 17 00:00:00 2001 From: hunjyeong Date: Sat, 30 Nov 2024 22:39:34 +0900 Subject: [PATCH 1/4] docs/group.md --- content/Basic Algebra/Basic Algebra.md | 2 +- content/Basic Algebra/Group.md | 24 +++++++++++++++++++++++- 2 files changed, 24 insertions(+), 2 deletions(-) diff --git a/content/Basic Algebra/Basic Algebra.md b/content/Basic Algebra/Basic Algebra.md index 3f99eba..5b357c0 100644 --- a/content/Basic Algebra/Basic Algebra.md +++ b/content/Basic Algebra/Basic Algebra.md @@ -24,4 +24,4 @@ - [[Discrete logarithm]] - [[ECDLP]] - [[Ideal]] -- [[Abelian Group]] + diff --git a/content/Basic Algebra/Group.md b/content/Basic Algebra/Group.md index 6778876..973acfa 100644 --- a/content/Basic Algebra/Group.md +++ b/content/Basic Algebra/Group.md @@ -8,4 +8,26 @@ Given a [[Binary Operation]] $*$ : $G \times G \rightarrow G$ on a set $G$, the # Example -- $GL_{2}(\mathbb{R})$ is the set of all invertible 2×2 matrices, and with matrix multiplication, it forms a group. \ No newline at end of file +- $GL_{2}(\mathbb{R})$ is the set of all invertible 2×2 matrices, and with matrix multiplication, it forms a group. + +
+ +**Additional Information** + +- 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 + +
+ +## 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. \ No newline at end of file From 1d31a60274d9dbe27bda3bdb67be6511af302861 Mon Sep 17 00:00:00 2001 From: hunjyeong Date: Sat, 30 Nov 2024 23:19:17 +0900 Subject: [PATCH 2/4] docs/group.md --- content/Basic Algebra/Group.md | 6 +++--- package-lock.json | 1 + 2 files changed, 4 insertions(+), 3 deletions(-) diff --git a/content/Basic Algebra/Group.md b/content/Basic Algebra/Group.md index 973acfa..f7d5a20 100644 --- a/content/Basic Algebra/Group.md +++ b/content/Basic Algebra/Group.md @@ -6,7 +6,7 @@ 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. @@ -21,11 +21,11 @@ Given a [[Binary Operation]] $*$ : $G \times G \rightarrow G$ on a set $G$, the
-## Abelian Group +# Abelian Group A group $G$ under $*$ is an abelian group if $*$ is commutative. -# Example +### Example - $(\mathbb{R}, +)$ is an abelian group. - $(\mathbb{R}^*, \cdot)$ is an abelian group. ($\mathbb{R}^* = \mathbb{R} \ {0}$) diff --git a/package-lock.json b/package-lock.json index 994714e..f2edf94 100644 --- a/package-lock.json +++ b/package-lock.json @@ -1341,6 +1341,7 @@ "version": "0.5.0", "resolved": "https://registry.npmjs.org/async-mutex/-/async-mutex-0.5.0.tgz", "integrity": "sha512-1A94B18jkJ3DYq284ohPxoXbfTA5HsQ7/Mf4DEhcyLx3Bz27Rh59iScbB6EPiP+B+joue6YCxcMXSbFC1tZKwA==", + "license": "MIT", "dependencies": { "tslib": "^2.4.0" } From 9697649f2ff6fa0b2eae6723eff07a525b97f79f Mon Sep 17 00:00:00 2001 From: hunjyeong Date: Sat, 30 Nov 2024 23:21:33 +0900 Subject: [PATCH 3/4] docs/group.md --- content/Basic Algebra/Group.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/content/Basic Algebra/Group.md b/content/Basic Algebra/Group.md index f7d5a20..d544457 100644 --- a/content/Basic Algebra/Group.md +++ b/content/Basic Algebra/Group.md @@ -16,8 +16,8 @@ Given a [[Binary Operation]] $*$ : $G \times G \rightarrow G$ on a set $G$, the - 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 +- 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. $a * a^{-1} = a^{-1} * a = e$
@@ -28,6 +28,6 @@ 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. +- $(\mathbb{R}^*, \cdot )$ 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. \ No newline at end of file From 5fb29e1926ba8891862d2b93ec4462536c35774d Mon Sep 17 00:00:00 2001 From: hunjyeong Date: Sun, 1 Dec 2024 16:03:12 +0900 Subject: [PATCH 4/4] docs/group.md --- content/Basic Algebra/Group.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/Basic Algebra/Group.md b/content/Basic Algebra/Group.md index d544457..200bb74 100644 --- a/content/Basic Algebra/Group.md +++ b/content/Basic Algebra/Group.md @@ -12,7 +12,7 @@ Given a [[Binary Operation]] $*$ : $G \times G \rightarrow G$ on a set $G$, the
-**Additional Information** +**Note** - Commutativity: For $ \forall a, b \in G$, $a * b = b * a$ - Associativity: For $ \forall a, b, c \in G$, $(a * b) * c = a * (b * c)$