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[algebraic structure] group (definition of group | basic properties of group | proof method of group | commutative group)

2022-07-03 03:10:00 【Programmer community】

List of articles

- - - The definition of group
    - Classification of groups
    - Proof method of group
    - Proof method of commutative group
    - Review of several episodes
    - Proof of group

The definition of group

Group Of Definition : One Non empty aggregate

$G$ in , If Defined One “ Multiplication ” operation , Satisfy the following four nature , that The Nonempty set

$G$ be called Group ;

1. Sealing property :
- 1> Symbolic representation : $\forall a,b \in G , a \times b = c \in G$
  $\forall a, b \in G, a \times b = c \in G$
- 2> Natural language description : Nonempty set $c c c It's also aggregate G G$
  $a, b$ Multiply , As a result,
  $G$ The elements in ;
2. Associative law :
- Symbolic representation : $\forall a,b, c \in G , a \times ( b \times c ) = (a \times b) \times c$
  $\forall a, b, c \in G, a \times (b \times c) = (a \times b) \times c$ ;
3. There are unit yuan :
- 1> Symbolic representation : $\exist e \in G, \forall a \in G, e \times a = a\times e = a$
  $\exists e \in G, \forall a \in G, e \times a = a \times e = a$
- 2> Natural language description : There is one. $a a a, perhaps And a a a Multiply, The result is a a a, amount to 11$
  $e$ , multiply
  $1$ ;
4. Yuan per person $a^{-1}$
$a^{- 1}$ :

Be careful :
This “ Multiplication ” Refers to the “ Multiplication ” , namely Of the elements in a collection Two element operation ;
G
×
G
G \times G
$G \times G$ The algebraic structure can be expressed as
(
G
,
⋅
)
( G , \cdot )
$(G, \cdot)$

Classification of groups

Group Of classification :

1. Exchange group ( Abel Group ) : Commutative law Established Group , be called Exchange group or Abel Group ;
2. Noncommutative groups ( Not Abel Group ) : Commutative law It doesn't work Group , be called Noncommutative groups or Not Abel Group ;
3. Group Of rank : Group $∣ G ∣ |G|$
$G$ The number of elements contained is called the order of the Group , Remember to do
$∣ G ∣$ ;
4. Finite group :
$∣ G ∣$ yes Limited , be called Finite group ;
5. Infinite group :
$∣ G ∣$ yes infinite , be called Infinite group ;

Proof method of group

Proof method of group : Given a aggregate

$G$ and Two element operation , Prove that the set is a group ;

1. Nonempty set : First of all The set is a non empty set ;
2. Prove closeness : aggregate in Any two elements Carry out operations Got The third element Must also be Collection ;
3. Prove the law of Association : Collection
$c$ The result of the operation is the same ;
4. Prove that it has unit yuan : There is a in the collection
$0$ ;
5. Prove its inverse : $a^{-1}a−1 perhaps a − 1 a^{-1}a−1 And a aa Carry out operations , As a result, e e$
$e$ Unit element ;

To meet the above
4
4
$4$ Conditions , You can prove that This collection It's a About this operation Group ;

Proof method of commutative group

Based on the proof method of group , Prove that its exchange law is established ;

Review of several episodes

Number set And Representation :

1. Integers :
$Z$ , A set of all integers , be called Set of integers ;
2. Positive integer : $Z^+,N^*,N^+$
$Z^{+}, N^{*}, N^{+}$ , A set of all positive integers , It is called the set of positive integers ;
3. Negtive integer : $Z^-$
$Z^{-}$ , A set of all negative integers , Called negative integer set ;
4. Non-negative integer :
$N$ , A set of all nonnegative integers , It is called a set of nonnegative integers ( or Set of natural numbers ) ;
5. Rational number :
$Q$ , All rational numbers Set of components , It is called a set of rational numbers ;
6. Set of real numbers :
$R$ , A set of all real numbers , It is called the set of real numbers ;
7. imaginary number :
$I$ , A set of all imaginary numbers , It is called the set of imaginary numbers ;
8. The plural :
$C$ , All real numbers and imaginary number Set of components , It is called a complex set ;

Rational number : It is generated by integer division , It can be expressed by scores , The decimal part is Co., LTD. or Infinitely recurring decimals ;
The set of real Numbers : Irrational numbers are generally generated by the square of positive integers , Real numbers correspond to points on the number axis one by one , Contains rational numbers and Irrational number , Irrational numbers are infinite non recurring decimals ;
imaginary number : Imaginary numbers are generally generated when the square is negative or the radical is negative , Imaginary numbers are divided into real parts or Imaginary part ;

Common superscripts in data sets usage :

1. Positive numbers : $^+$
$^{+}$ Indicates that all elements in the data set are Positive numbers ;
2. negative : $^-$
$^{-}$ Indicates that all elements in the data set are negative ;
3. To eliminate $*^*^{*} It means to eliminate the elements on the data set 00 0;$
$0$ Elements :

R
∗
R^*
$R^{*}$ Means to eliminate Set of real numbers
R
R
$R$ Medium Elements
0
0
$0$ ,
R
∗
=
R
∖
{
0
}
=
R
−
∪
R
+
=
(
−
∞
,
0
)
∪
(
0
,
+
∞
)
R^* = R \setminus \{0\} = R^- \cup R^+ = (- \infty , 0) \cup (0,+ \infty)
$R^{*} = R ∖ {0} = R^{-} \cup R^{+} = (- \infty, 0) \cup (0, + \infty)$

Proof of group

subject : Prove all rational numbers About Multiplication Form a group ;

Method of proof : Given a aggregate

$G$ and Two element operation , Prove that the set is a group ;

1. Nonempty set : First of all The set is a non empty set ;
2. Prove closeness : aggregate in Any two elements Carry out operations Got The third element Must also be Collection ;
3. Prove the law of Association : Collection
$c$ The result of the operation is the same ;
4. Prove that it has unit yuan : There is a in the collection
$0$ ;
5. Prove its inverse : $a^{-1}a−1 perhaps a − 1 a^{-1}a−1 And a aa Carry out operations , As a result, e e$
$e$ Unit element ;

To meet the above
4
4
$4$ Conditions , You can prove that This collection It's a About this operation Group ;

prove :

① Sealing property : Rational number Multiply It must also be rational , Satisfy closure ;
② Associative law :

$3$ individual arbitrarily Rational number Multiply , Obviously Satisfy Associative ;
③ Proof unit: Yuan : There is

e=1

$e = 1$ , Rational number multiply 1 perhaps 1 multiply Rational number , Are equal to the rational number , It indicates that the unit element exists ;
④ Proof inverse

−

a^{-1}

$a^{- 1}$ The existence of : Any element in the set