What kind of background? Its inventor is Evariste Galois, yes, Galois. /kloc-before the age of 0/2, self-study at home. /kloc-at the age of 0/4, he began to get tired of other subjects and was only interested in mathematics. /kloc-started reading Lagrange's thesis at the age of 0/5. 17 years old, published his first paper. In the same year, he tried to be admitted to the Paris Institute of Technology and was rejected (it is said that he skipped too many reasoning steps in the interview and confused the examiner. Finally, he can't stand the slow pace of the examiner. In a rage, he grabbed a rag to clean the blackboard, threw it at the examiner and hit it directly. /kloc-at the age of 0/8, he was expelled from Paris Teachers College for publishing an open letter criticizing the headmaster. 19 years old, arrested several times for participating in political activities. At the age of 20, he took part in a duel (probably because of love) and was shot in the abdomen.
Portrait of galois. Image source: Wikipedia
What is the definition? Strictly speaking, it will be annoying, but the basic principle is simple: first, you have to have a bunch of things (sets), and then you can get a result by putting any two of them together in some way (operation). A set, plus a binary operation, that's it.
For example. We deal with a very common group every day, and mathematicians give it a name "integer addition group": integers are a pile of things (sets) we have; Addition is the way we put these things together. Just try it. If you find two integers at random, you can add them up and there must be a result.
therefore ...
If the WeChat group is a real group (1)
The "collection" of WeChat group seems to be a collection of group members; An element is a person. It requires a binary operation, which can be called "interaction". According to the naming method just now, this is a "WeChat member interactive group", and any two group members must be able to interact together (please don't associate too much).
No problem here, but:
Group operation is particular.
Although you can build a group as long as you have a set and an operation, this operation does not qualify for any operation. Specifically, this operation should satisfy four "group axioms": closeness, associative law, unit element and inverse element.
Closed: No matter which two group members you take out, you will definitely get group members after the operation, and it is impossible to run out of the group. For example, if you add any two integers, you still have to get an integer.
Association rule: if you want to calculate three members, it doesn't matter which two you calculate first, and the results are the same. For example, (1+2)+3 = 1+(2+3).
Unit element: One member must not change another member after the operation. For example, 0: 0+5 = 5+0 = 5 of the integer addition group.
Inverse element: any member must have its own "inverse"-it can be changed back to the unit element after operation with its inverse element. For example, in the integer addition group, there is -7: 7+(-7) = (-7)+7 = 0 for 7.
So:
If the WeChat group is a real group (2)
Applying these four group axioms to WeChat group, we will get the following results:
Closure: Any two group members interact, and the result must still be a group member.
Law of Association: When three members interact, it doesn't matter which one is the first. (Interaction is a binary operation, so the three cannot interact at the same time. )
Company element: there must be a group member, which can be called group owner. When the group owner interacts with any member, the result is still that member. (It can be proved that a WeChat group has one and only one group owner. )
Inverse element: for any group member, there must be another member, and the result of their interaction is the group owner.
Here, we might as well set that every time two members "interact", the result must be @ to a certain group member. If there is no @ or the result of the same two people @ is different every time, it is not the kind of interaction we care about.
Just like this.
There can be another structure in the group.
In a group, some elements will form a small circle by themselves. It's not that they don't communicate with the outside world, but there is no doubt that they like to hold groups: the results of the elements in the small circle are still in this small circle, and their inverse elements are also in the small circle. In short, this small circle has also formed a group for original operation. Such a small circle is called a subgroup of a group.
Some subgroups are more special than others. They are not only a group, but also a group if the original group is "split". Such subgroups are called normal subgroups, and the groups obtained by dividing the original groups are called quotient groups. This division is not exactly the same as the division in digital operation, and can be regarded as a way to divide small circles.
If the WeChat group is a real group (3)
Wechat groups do not necessarily have subgroups. But if there is, there will be such a situation: members of a small circle in the group can interact with others, but the interaction of insiders will eventually reach an insider.
Since this small circle conforms to the definition of group, they can form a new group independently. In fact, they may have done so, and you, as an outsider, don't know! hahaha.
A WeChat group will also add people and kick people. But because one of the two elements of a group is a given set, every time someone is added and kicked, the group actually becomes a new group. In this sense, you can't step into the same WeChat group twice.
Why do you have to pay so much attention to getting a group?
As a mathematical concept, "group" was invented without any external compulsion. Mathematicians are not stupid. The purpose of inventing and defining it like this must be because it is useful.
Indeed, group is one of the most useful basic concepts in modern mathematics. When Galois wrote down the term "group theory" at that time, he mainly considered the problem of solving equations more than five times, but today its use far exceeds that field, because later we learned that the biggest use of group theory is to study "symmetry"; Group theory can come in handy for everything with symmetry.
The meaning of symmetry here is even wider than that of everyday language. For mathematicians, as long as something is unchanged after transformation, it is symmetrical. Geometry can of course be symmetrical: a circle is still a circle after rotating left and right, and it is still a circle after rotating 180 degrees, so it is symmetrical under these two transformations. But symmetry can also be applied to non-geometric abstract concepts: for example, the function f (x, y, z) = x 2+y 2+z 2 remains unchanged no matter how the positions of x, y and z change; Or sin(t), replacing t with t+2π, is also unchanged. They also have corresponding symmetry.
The most amazing thing about symmetry is that it corresponds to the conservation in the physical world. For example, the laws of physics will not change with the passage of time, in other words, they are symmetrical under the time transformation; And this symmetry can directly deduce one of the most important laws in physics: the conservation of energy. The laws of physics will not change with the position of space, and this symmetry can lead to another equally critical law: the conservation of momentum. Every physical conserved quantity must be accompanied by mathematical symmetry, which was discovered by emmy noether, one of the greatest mathematicians in the 20th century.
Amy Norther is a master in the field of abstract algebra; Her Nott theorem is one of the mathematical foundations of Einstein's general theory of relativity. Image source: huffingtonpost.com
In addition, modern particle physics relies entirely on group theory. All kinds of new particles can be neatly classified into the standard model because of the study of symmetry; In fact, quite a few new particles were first predicted by group theory and then discovered by experiments.
Chemistry and biology are also inseparable from group theory-there are too many symmetries in molecules and crystals to deal with their structures and behaviors without group theory.
Even the Rubik's cube is a group: the small squares in the Rubik's cube can be regarded as elements of mass, rotating the Rubik's cube is equivalent to operation, and the Rubik's cube formula can also be obtained by group theory. Image source: Wikipedia
If the WeChat group is a real group (summary)
It can be said that every specific group has always existed in this world, just waiting for people to discover it. So your WeChat group may already be a group! Quickly check the requirements list:
It should have a bunch of given members.
It must have a given binary operation (for example, a chat between two people ends with @ a member).
It should be closed (people outside the group can't go)
It should have a law of association (the order of interaction doesn't matter)
It must have a unit element (the interaction between the group owner and anyone must end with @ this person).
It must have an inverse element (for anyone, there is a member, and the interaction between two people is bound to be noisy (fog) and the @ group owner will make a ruling)
If these conditions are met, congratulations, a hidden and unknown group has been discovered by you! If these conditions are not met, congratulations. We have proved mathematically that this is not a group at all. What else can we do?