From 9984325e666d9c3175f2fc083f2c53a377a0277b Mon Sep 17 00:00:00 2001 From: Cminorwhy <112066107+Cminorwhy@users.noreply.github.com> Date: Tue, 19 Dec 2023 20:02:24 +0000 Subject: [PATCH] small motify --- .../1_The_Algebriac_Structure_of_Groups.md | 102 +++++++++--------- 1 file changed, 52 insertions(+), 50 deletions(-) diff --git a/study/Imperial_mathematics/year_2/Groups_and_Rings/Part_1_Groups/1_The_Algebriac_Structure_of_Groups.md b/study/Imperial_mathematics/year_2/Groups_and_Rings/Part_1_Groups/1_The_Algebriac_Structure_of_Groups.md index 860bdf4cb..b4b6bc2b8 100644 --- a/study/Imperial_mathematics/year_2/Groups_and_Rings/Part_1_Groups/1_The_Algebriac_Structure_of_Groups.md +++ b/study/Imperial_mathematics/year_2/Groups_and_Rings/Part_1_Groups/1_The_Algebriac_Structure_of_Groups.md @@ -52,7 +52,7 @@ $$\gdef\abs#1{\vert #1 \vert}$$ > - `Inverse` $$\imply$$ the uniqueness of $$a^{-1}$$. -- By contradiction. -#### $$\bluetext{Proposition 1.1 Cancellation Law}$$ +#### $$\bluetext{Theorem 1.1 Cancellation Law}$$ - $$G$$ is a group. @@ -72,7 +72,6 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the - `Commutivity:` $$\forall a, b \in G, a b = b a$$. - #### $$\bluetext{Definition (Order of a Group)}$$ - $$G$$ is a group. @@ -136,7 +135,7 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the $$\def$$ the **right coset** of $$H$$ in $$G$$ by some $$g \in G$$ is $$H g := \set{h g \mid h \in H}$$. -> `In same left/right cosets <=> substraction from left/right gives a element of subgroup:` +> - `In same left/right cosets <=> substraction from left/right gives a element of subgroup:` > > - $$\forall a, b \in G$$, $$a$$ and $$b$$ are in some same left cosets of $$H$$ in $$G$$ $$\ifif$$ $$a^{-1} b \in H$$. > @@ -150,7 +149,7 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the > > --- > -> `For every element in a left/right coset <=> generates the same left/right coset:` +> - `For every element in a left/right coset <=> generates the same left/right coset:` > > - $$\forall a, b \in G$$, $$aH = bH$$ $$\ifif$$ $$a \in bH$$. --- > @@ -164,13 +163,16 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the > > --- > -> `All left/right cosets form a partition of the whole group:`$$\forall a \in G$$, $$\exists! g \in G$$, s.t. $$a \in gH$$ and $$\exists! g \in G$$, s.t. $$a \in Hg$$. +> - `All left/right cosets form a partition of the whole group:`$$\forall a \in G$$, $$\exists! g \in G$$, s.t. $$a \in gH$$ and $$\exists! g \in G$$, s.t. $$a \in Hg$$. > > --- -> `All left/right cosets have the same cardinality:` +> - `All left/right cosets have the same cardinality:` > > - $$\forall a \in G$$, $$\abs{aH} = \abs{Ha} = \abs{H}$$. -- Check that the map $$h \mapsto ah(ha)$$ is a bijection from $$H$$ to $$aH(Ha)$$. + +> For the following statments about **"could be seen"**, actually it means that there is a **group isomorphism** between the two groups, which we will discuss later and the relation $=$ dose not mean that the equlity holds in the set theory sense. We just use $=$ to denote the **isomorphism**. (Since we have not define the isomorphism yet, we will not use $=$ to denote the isomorphism in the following.) + ### $$\bluetext{1.2 Some Abstract Groups}$$ ### $$\bluetext{1.2.1 Product Groups}$$ @@ -185,17 +187,15 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the `Well-definedness:` Check group axioms (`associativity`, `identity`, `inverse`) for $$G \times H$$. -> `Trivial product group:` +> - `Trivial product group:` > -> - $$\set{(g, e) \mid g \in G} \leq G \times H$$ and $$\set{(e, h) \mid h \in H} \leq G \times H$$. -- Check `identity`, `inverse` and `closure`. -> -> - $$\set{(e, h) \mid h \in H} \leq G \times H$$ and $$\set{(g, e) \mid g \in G} \leq G \times H$$. -- Check `identity`, `inverse` and `closure`. +> $$\set{(g, e) \mid g \in G} \leq G \times H$$ and $$\set{(e, h) \mid h \in H} \leq G \times H$$. -- Check `identity`, `inverse` and `closure`. > > --- > -> `The product group is abelian <=> the two groups are abelian:` +> - `The product group is abelian <=> the two groups are abelian:` > -> - $$G \times H$$ is abelian $$\ifif$$ $$G$$ and $$H$$ are abelian. -- By the definition of product group. +> $$G \times H$$ is abelian $$\ifif$$ $$G$$ and $$H$$ are abelian. -- By the definition of product group. ### $$\bluetext{1.2.2 Symmetric Groups and Alternating Groups}$$ @@ -209,15 +209,21 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the > For the followng examples, let $$X = \{1, 2, 3, 4, 5\}$$. > -> - `Cycle notation:` The permutation $$f$$ could be written as **a product of cycle notation**, e.g. $$(1 2 3)(4 5)$$ means $$1 \to 2 \to 3 \to 1$$ and $$4 \to 5 \to 4$$, where each pair of parentheses represents a **cycle**. +> - `Cycle notation:` +> +> The permutation $$f$$ could be written as **a product of cycle notation**, e.g. $$(1 2 3)(4 5)$$ means $$1 \to 2 \to 3 \to 1$$ and $$4 \to 5 \to 4$$, where each pair of parentheses represents a **cycle**. > > --- > -> - `Two-row notation:`The permutation $$f$$ could also be written in **two-row notation**,e.g. $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 1 & 5 & 4 \end{pmatrix}$$, where the first row represents the elements of $$X$$ and the second row represents the images of the elements of $$X$$. +> - `Two-row notation:` +> +> The permutation $$f$$ could also be written in **two-row notation**,e.g. $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 1 & 5 & 4 \end{pmatrix}$$, where the first row represents the elements of $$X$$ and the second row represents the images of the elements of $$X$$. > > --- > -> - `Matrix notation:` The permutation $$f$$ could be written as a **matrix**, e.g. $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ 1 \\ 5 \\ 4 \end{pmatrix}$$, if we use vector to represent the elements of $$X$$ by the order of $$X$$. Note that we are not treating the set consists of "1, 2, 3, 4, 5" as a vector space (Since we have not verify it.), but we are using the matrix to represent the permutation $$f$$. Since a permutation could be seen as a linear transformation, we could use matrix to represent it. +> - `Matrix notation:` +> +> The permutation $$f$$ could be written as a **matrix**, e.g. $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ 1 \\ 5 \\ 4 \end{pmatrix}$$, if we use vector to represent the elements of $$X$$ by the order of $$X$$. Note that we are not treating the set consists of "1, 2, 3, 4, 5" as a vector space (Since we have not verify it.), but we are using the matrix to represent the permutation $$f$$. Since a permutation could be seen as a linear transformation, we could use matrix to represent it. #### $$\bluetext{Definition (Symmetric Group)}$$ @@ -229,9 +235,9 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the `Well-definedness:` Check group axioms (`associativity`, `identity`, `inverse`) for $$S_n$$. -> `Cardinality of symmetric group:` +> - `Cardinality of symmetric group:` > -> $$\abs{S_n} = n!$$ -- To count the number of permutations of $$X$$, we can first choose the image of $$1$$ from $$n$$ elements, then choose the image of $$2$$ from $$n - 1$$ elements, and so on. +> $$\abs{S_n} = n!$$ -- To count the number of permutations of $$X$$, we can first choose the image of $$1$$ from $$n$$ elements, then choose the image of $$2$$ from $$n - 1$$ elements, and so on. #### $$\bluetext{Definition (Sign of a Permutation)}$$ @@ -274,11 +280,13 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the - $$\def$$ $$g$$ is a **generator** of $$C_n$$ $$\ifif$$ $$G = \langle g \rangle$$. -> - The generator of a cyclic group is not unique. -- For example, $$1$$ and $$-1$$ are both generators of $$(\Z, +) = \langle 1 \rangle = \langle -1 \rangle$$. +> - `The generator of a cyclic group is not unique.` -- For example, $$1$$ and $$-1$$ are both generators of $$(\Z, +) = \langle 1 \rangle = \langle -1 \rangle$$. > > --- > -> - `Cyclic group is abelian:` $$C_n$$ is cyclic $$\imply$$ $$\forall g^i, g^j \in G, g^i g^j = g^j g^i$$ -- By the definition of cyclic group. But, the converse is not true. -- For example, $$(\Q, +)$$ is abelian but not cyclic. +> - `Cyclic group is abelian:` +> +> $$C_n$$ is cyclic $$\imply$$ $$\forall g^i, g^j \in G, g^i g^j = g^j g^i$$ -- By the definition of cyclic group. But, the converse is not true. -- For example, $$(\Q, +)$$ is abelian but not cyclic. > > --- > @@ -313,17 +321,15 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the `Well-definedness:` Check group axioms (`associativity`, `identity`, `inverse`) for $$GL(n, \F)$$. -> For the following statments about "could be seen", actually it means that there is a **group isomorphism** between the two groups, which we will discuss later. - -> `Symmetric group is a subgroup of general linear group:` +> - `Symmetric group is a subgroup of general linear group:` > -> - The symmetric group $$S_n$$ could be seen as a subgroup of $$GL(n, \F)$$ by identifying the permutation $$f$$ with the matrix $$A$$ whose $$i$$-th column is the $$f(i)$$-th standard basis vector. -- For example, $$f = (1 2 3) \in S_3$$ could be identified with $$A = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \in GL(3, \F)$$. +> The symmetric group $$S_n$$ could be seen as a subgroup of $$GL(n, \F)$$ by identifying the permutation $$f$$ with the matrix $$A$$ whose $$i$$-th column is the $$f(i)$$-th standard basis vector. -- For example, $$f = (1 2 3) \in S_3$$ could be identified with $$A = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \in GL(3, \F)$$. > > --- > > `Cyclic group is a subgroup of general linear group:` > -> - $$C_n$$ could be seen as a subgroup of $$GL(1, \F)$$ by using the rotational matrix $$A = \begin{pmatrix} \cos \frac{2 \pi}{n} & -\sin \frac{2 \pi}{n} \\ \sin \frac{2 \pi}{n} & \cos \frac{2 \pi}{n} \end{pmatrix} \in GL(2, \F)$$ as the generator. +> $$C_n$$ could be seen as a subgroup of $$GL(1, \F)$$ by using the rotational matrix $$A = \begin{pmatrix} \cos \frac{2 \pi}{n} & -\sin \frac{2 \pi}{n} \\ \sin \frac{2 \pi}{n} & \cos \frac{2 \pi}{n} \end{pmatrix} \in GL(2, \F)$$ as the generator. #### $$\bluetext{Definition (Special Linear Group)}$$ @@ -365,8 +371,6 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the ### $$\bluetext{1.3.2 Klein Four-group and Dihedral Group}$$ - - #### $$\bluetext{Definition (Dihedral Group)}$$ - $$n \in \N^+$$. @@ -379,30 +383,6 @@ $$\def$$ $$G$$ is an **abelian group** $$\ifif$$ $$G$$ is a group satisfying the `Well-definedness:` Check group axioms (`associativity`, `identity`, `inverse`) for $$D_{n}$$. -> `Dihedral group is a subgroup of orthogonal group:` -> -> - $$D_{n} \leq O(2, \R)$$. -- By the definition of dihedral group. The dihedral is a subgroup that consists the Euclidean distance-preserving transformations of the plane, which includes rotations and reflections. -> -> --- -> -> `Dihedral group is a subgroup of symmetric group:` -> -> - $$D_{n} \leq S_n$$. -- Since rotation and reflection could be seen as a permutation of the vertices of a regular $$n$$-gon if we identify the vertices with the elements of $$X = \{1, \dots, n\}$$. -> -> --- -> -> `Cardinality of dihedral group:` -> -> - $$\abs{D_{n}} = 2n$$. -- By the definition of dihedral group. -> -> --- -> -> `Dihedral group is non-abelian if n >= 3:` -> -> - $$D_{n}$$ is non-abelian if $$n \geq 3$$. -- When $$n = 1$$, $$D_{1} = \set{e, (1, 2)} = C_2$$, which is abelian. When $$n = 2$$, $$D_{2} = \set{e, r, s, rs} = K_4$$, which is abelian (Introduced in the following.). -> ). When $$n \geq 3$$, $$D_{2n}$$ is non-abelian. -- A conterexample: when $$n = 3$$, $$rs = (1 2)$$ and $$sr = (1 3)$$, so $$rs \neq sr$$. - - #### $$\bluetext{Definition: Klein Four-group}$$ $$\def$$ $$K_4 := \set{e, a, b, c}$$ with the group operation defined by the following table: @@ -423,6 +403,28 @@ $$\def$$ $$K_4 := \set{e, a, b, c}$$ with the group operation defined by the fol > > - $$K_4 = D_4 \leq S_4$$. -- Write $$K_4$$ in permutation form, i.e. $$K_4 = \set{e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}$$. +> - `Dihedral group is a subgroup of orthogonal group:` +> +> $$D_{n} \leq O(2, \R)$$. -- By the definition of dihedral group. The dihedral is a subgroup that consists the Euclidean distance-preserving transformations of the plane, which includes rotations and reflections. +> +> --- +> +> - `Dihedral group is a subgroup of symmetric group:` +> +> $$D_{n} \leq S_n$$. -- Since rotation and reflection could be seen as a permutation of the vertices of a regular $$n$$-gon if we identify the vertices with the elements of $$X = \{1, \dots, n\}$$. +> +> --- +> +> - `Cardinality of dihedral group:` +> +> $$\abs{D_{n}} = 2n$$. -- By the definition of dihedral group. +> +> --- +> +> - `Dihedral group is non-abelian if n >= 3:` +> +> $$D_{n}$$ is non-abelian if $$n \geq 3$$. -- When $$n = 1$$, $$D_{1} = \set{e, (1, 2)} = C_2$$, which is abelian. When $$n = 2$$, $$D_{2} = \set{e, r, s, rs} = K_4$$, which is abelian (Introduced in the following.) When $$n \geq 3$$, $$D_{n}$$ is non-abelian. -- A conterexample: when $$n = 3$$, $$rs = (1 2)$$ and $$sr = (1 3)$$, so $$rs \neq sr$$. + ### $$\bluetext{1.4 Groups Defined on Integers Mod n}$$