離散數學群論

半群 (Semigroup)

An algebraic structure (G, *) is said to be a semigroup. If the binary operation * is associated in G i.e. if (a*b) *c = a *(b*c) a,b,c e G. For example, the set of N of all natural number is semigroup with respect to the operation of addition of natural number.

代數結構(G，*)被稱為半群。 如果二進制運算*與G關聯，即(a * b)* c = a *(b * c)a，b，ce G。 例如，相對于自然數的加法運算，所有自然數的N的集合是半群。

Obviously, addition is an associative operation on N. similarly, the algebraic structure (N, .)(I, +) and (R, +) are also semigroup.

顯然，加法是對N的關聯運算。 同樣，代數結構(N，。)(I，+)和(R，+)也是半群。

單體 (Monoid)

A group which shows property of an identity element with respect to the operation * is called a monoid. In other words, we can say that an algebraic system (M,*) is called a monoid if x, y, z E M.

顯示關于操作*的標識元素的屬性的組稱為monoid。 換句話說，如果x，y，z EM ，我們可以說一個代數系統(M，*)被稱為一個等式 。

(x *y) * z = x * (y * z)

(x * y)* z = x *(y * z)

And there exists an elements e E M such that for any x E M

并且存在一個元素EM ，對于任何x EM

e * x = x * e = x where e is called identity element.

e * x = x * e = x其中e稱為身份元素。

關閉屬性 (Closure property)

The operation + is closed since the sum of two natural number is a natural number.

由于兩個自然數之和是自然數，所以運算+是閉合的。

關聯財產 (Associative property)

The operation + is an associative property since we have (a+b) + c = a + (b+c) a, b, c E I.

由于我們具有(a + b)+ c = a +(b + c)a，b，c EI，因此運算+是一種關聯性質。

身分識別 (Identity)

There exist an identity element in a set I with respect to the operation +. The element 0 is an identity element with respect to the operation since the operation + is a closed, associative and there exists an identity. Since the operation + is a closed associative and there exists an identity. Hence the algebraic system ( I, +) is a monoid.

關于操作+ ，在集合I中存在一個標識元素。 元素0是關于操作的標識元素，因為操作+是封閉的，關聯的并且存在一個標識。 由于操作+是封閉的關聯，因此存在一個標識。 因此，代數系統(I，+)是一個齊半群 。

組 (Group)

A system consisting of a non-empty set G of element a, b, c etc with the operation is said to be group provided the following postulates are satisfied:

如果滿足以下假設，則一個由元素a，b，c等組成的非空集G組成的系統將被稱為組。

1. Closure property

1.關閉屬性

    For all a, b E G => a, b E G
i.e G is closed under the operation ‘.’

2. Associativity

2.關聯性

    (a,b).c = a.(b.c) a, b, c E G.
i.e the binary operation ‘.’ Over g is associative.

3. Existence of identity

3.身份的存在

    There exits an unique element in G. Such that e.a = a = a.e 
for every a E G. This  element e is called the identity.

4. Existence of inverse

4.逆的存在

    For each a E G , there exists an element a^-1 E G 
such that a. a^-1 = e = a^-1.a
the element a^-1 is called the inverse of a .

交換組 (Commutative Group)

A group G is said to be abelian or commutative if in addition to the above four postulates the following postulate is also satisfied.

如果除上述四個假設外，還滿足以下假設，則稱G組為阿貝爾或交換性的。

5. Commutativity

5.可交換性

    a.b = b.a for every a, b E G.

循環群 (Cyclic Group)

A group G is called cyclic. If for some aEG, every element xEG is of the form a^n. where n is some integer. Symbolically we write G = {a^n : n E I}. The single element a is called a generator of G and as the cyclic group is generated by a single element, so the cyclic group is also called monogenic.

組G稱為循環的。 如果對于某些aEG ，每個元素xEG的形式都是a ^ n 。 其中n是一些整數。 象征性地，我們寫G = {a ^ n：n EI} 。 單個元素a稱為G的生成器，并且由于環狀基團是由單個元素生成的，因此環狀基團也稱為單基因 。

亞組 (Subgroup)

A non- empty subset H of a set group G is said to be a subgroup of G, if H is stable for the composition * and (H, *) is a group. The additive group of even integer is a subgroup of the additive group of all integer.

一組群G的一個非空真子集H被表示為G的一個子群，如果H是穩定該組合物*和(H，*)是一組。 偶數整數的加法組是所有整數的加法組的子組。

翻譯自: https://www.includehelp.com/basics/group-theory-and-their-type-in-discrete mathematics.aspx

離散數學群論