1) any a, b∈G, then a+b ∈ g;
2) where a, b, c∈G, then (a+b)+c = a+(b+c);
3) there is 0, and for any a∈G, there is a+0 = 0+a = a;
4) For any a∈G, there is -a, so a+(-a)=(-a)+a=0.
Question 2: What are the differences and connections among groups, fields and rings in mathematics? (1) group: A binary operation is defined on *** G, which satisfies the following four conditions:
End. 2. The law of association. 3. What is included? 4. There are opposites.
Then the algebraic structure (g, *) formed by this * * and binary operation is called a group.
(2) Abel group: a group whose binary operation also satisfies the commutative law. So Abel group, also called commutative group, is a special group. Binary operations are recorded as "+"
(3) Semigroups: Binary operations defined on * * * satisfy the first two conditions:
1. End. 2. The law of association. A group must be a semigroup, but a semigroup is not necessarily a group. )
With the above definition, let's take a look at what rings and domains are.
(4) Ring: Let *** R define two binary operations "+"and "*" and satisfy.
1.(R,+) is an Abelian group.
2.(R, *) is a semigroup.
3. The two operations meet the allocation rate, a * (b+c) = a * b+a * c.
Then the algebraic structure composed of *** R and two binary operations is called a ring.
(5) Domain: A semigroup structure in a ring is called a domain if it satisfies the inclusion law and the commutative law. Visible domain is a special kind of ring.
To sum up: the biggest concept is semigroup, group is a subset of semigroup and Abel group is a subset of group. The ring is modified on the basis of Abel group, that is, a binary operation is added to make * * * form a semigroup, and the two operations satisfy the distribution rate mentioned above. The last field is a subset of rings, and binary operations that need to be added must also satisfy the inclusion law and the commutative law.
Question 3: What's a good QQ group for math? Recommend a ... search-search group-search by conditions. You can find many groups you want by entering keywords and searching mathematically. The red one is the senior group. Good luck.
Question 4: What are the differences and connections among groups, fields and rings in mathematics? This is the content of abstract algebra:
* * * is a basic concept, equivalent to a class/a pile/all/... You should understand, don't say it again.
A group is a special set on which an operation can be defined (usually called "multiplication", but not necessarily related to the multiplication of numbers). It should be closed/combinable/have a unit element (similar to multiplication 1/ plus 0)/ have an inverse element (similar to multiplication reciprocal/addition inverse). ...
For example, positive rational numbers are multiplication groups, non-zero rational numbers are multiplication groups, and integer sets are grouped under addition.
Note that a group does not need a commutative law. If the commutative law is satisfied, it is called Abel group (or additive group).
The requirements of rings and domains are even higher. I don't need to talk to you in the abstract, just discuss it within the scope of numbers:
The set of closed numbers under addition/subtraction/multiplication is a number ring. If the number ring is also closed under division, it is called the number field.
All multiples of a number (including negative numbers) form a field ring, the rational number set is the smallest number field, and the real number set/complex number set is also a number field.
For more in-depth contents, please refer to college textbooks, abstract algebra/modern algebra, etc. ......
Question 5: Is Tianjin enough for one day? Main attractions in Tianjin:
About one day in Guangdong Club, Guwen Street and Water Park.
It takes about a day to mount the Huangyaguan Great Wall in Panshan.
It takes about a day for Dagubao Beach in Tanggu, Tianjin.
Zhou Deng Memorial Hall, Dabei Temple, Heather Food Street, Huo Yuanjia's former residence and Liang Qichao's former residence in the cemetery for about one day.