<Discrete Mathematics> Algebra System - group, semigroup

------ defined operations and properties

Let S be a non-empty, the mapping f: Sn-> S referred to as a n-ary operations on S. Suppose "•" is a binary operation defined on a set S. If:

1. ∀x, y∈S, x • y∈S, called "•" on the S is closed in.
2. ∀x, y∈S, x • y = y • x, called "•" on the S are exchangeable in.
3. ∀x, y, z∈S, x • (y • z) = (x • y) • z, called "•" is in S can bind to.
4. ∀x∈S, x • x = x, called "•" is on the S idempotent of.

 

◆ ☉ provided and define two binary operations simultaneously on the S, if

1. ∀x, y, zs, x☉ ( y ◆ z) = (x☉y) ◆ (x☉z) and y☉ (z ◆ x) = ( y☉x) ◆ (z☉x), called calculation About is assignable in.
2. ◆ ☉ and operation are interchangeable, and ∀x, yS, x ◆ (x☉y ) = x and x☉ (x ◆ y) = x , and ◆ is said to meet the operational ☉ absorption law .

 

------ algebraic system specific and element definitions

A non-empty set S is defined, along with a number of the operation on the S f1, F2, ..., fk called a system consisting of an algebraic system , referred to as <S, f1, f2, ... , fk>.

 

Specific elements defined algebraically system:
1. The identity element (unit cell) : If ∃eS such Xs, e • x = e • x = x, e is the identity element called algebraic systems.

2. Nil : If S ∃θS such that for any element x, there θ • x = x • θ = θ, [theta] is the algebraic system called the zero element.

3. Idempotents : If ∃a S, so that a • a = a, is a system called idempotent.

4. inverse : a • b = b • a = e, a and b are the inverse of each other.

- identity element and zero dollars is unique, each element if it has an inverse element inverse unique.

 

------group

For algebraic system <S, •>, if

1. "•" closed operations, for the wide group .

2. "•" is a closed operation, but also can be combined operations, compared with half the group .

3. "•" is a closed operation, but also can be combined operations, there is identity element, and each element has an inverse element, for the group .

 

Substitution of all provided on a set of n elements A configuration set S _n .

S _n- two permutation is a permutation on the composite remains in A, so that operation is closed;

Since the composite function is bound, it can be replaced with a composite is bonded; 

S _n- presence unitary permutation π = (1), so that any replacement of both σ • π = π • σ = σ, and thus π = (1) is a unitary element;

Each element in the x-y displacement becomes, the inverse permutation element put into x y, so that each has an inverse permutation;

We <S, •> called n symmetric group .

 

 

In both cases subgroups are:

设<G,*>和<S,*>都是群，若S是G的非空子集，则称S是G的子群。

设<G,*>是群，a ϵ G，记S={ aⁿ | n ϵ Z }，则<S,*>是<G,*>的子群。

（其他的定义也都可，满足第一条就行）

 

 

如果<S , •>是群，且运算满足交换律，则称<S , •>为可交换群。

<S , •>为可交换群   ↔ 对任意a，b ϵ G，都有( a • b )²=a² • b²

 

 

 

如果<S , •>是群，且其中存在一个元a使得群可由a生成，即G=(a)。则称G为循环群，a为G的一个生成元。称使得aⁿ=e的最小正整数n为元素a的周期。

在此基础上有三条推断可以直接使用：

1. a^m=e   ↔     n|m
2. aⁱ=a^j    ↔    n|(i-j)
3. 由a生成的子群恰有n个元素，即(a) = {e,a,a²,…,a^n-1}

 

拉格朗日定理

群G中子群H的所有左右陪集都是等势的；

n阶群<G，*>的任何子群<H，*>的阶必是n的因子；

n元群G中任何元素的周期必是n的因子。

——正规子群

设<H,*>是群<G,*>的一个子群。如果对于任何a ϵ G，aH=Ha 或 aHa^-1 ⊆ H，则称H是G的正规子群（或不变子群）。

——商群

设<H,*>是群<G,*>的一个正规子群，G/H表示G的所有陪集的集合，则<G/H，•>是一个群，称为商群。“•”定义为∀aH,bH ϵ G/H，aH•bH = (a*b)H 。

 

群的同态，群的同构

设<S,*>和<T,•> 是两个二元代数系统，

如果存在映射f：S→T，使得对任意a1,a2ϵS，f(a1*a2)=f(a1)•f(a2)，则称S，T同态，当f是双射时称f为同构映射。

设f是群<G,*>到<H,*>的同态映射，e‘是H的幺元，记Ker（f）={x|xϵG ∧ f(x)=e'}，Ker（f）称为f的同态核。 Ker（f）是G的正规子群。