Quick foundation of group theory (5): generators, Kelley graphs, orbits, cyclic graphs, and "dimensions" of groups?

Catalog

0. Preface

1. The generator

2. The drawing of Kelley diagram

3. Does the group have the concept of dimension ？

4. Track and cycle diagram

4.1 orbital

4.2 Cycle chart

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0. Preface

        Hobbies: Xiaobai's self-study notes on group theory . No purpose , Learning for learning . Organize and record along your own ideas in a way you can understand （ Imitate Kobayashi's copywriting and Mathematics ）, Not rigorous and complete , But for logical coherence .

         In this article, we will discuss the generators of groups and the drawing of Kelley graphs , On the basis of Kelley diagram, we further discuss orbit and cycle diagram .

         The generator of a group has some similarities with the basis of a linear space , Based on this , It's natural to ask a question ： Linear space has the concept of dimension , So does group have a concept similar to dimension ？

1. The generator

         The generator of a group refers to a basic set of elements of a group , The combinatorial operation based on this set of elements can generate all other elements .

         example 1. For example, cyclic groups , Element set {1} It is a set of generators . Based on elements 1, You can generate all other elements , for instance ,2=(1+1)%6, 3=(1+1+1)%6,0=(1+1+1+1+1+1)%6, wait .

        example 2. 《 Group theory color chart version 》2.2 In the group describing the symmetry of the flip of rectangular cards , Flip horizontally and flip vertically （ effect ） It forms two generators of this group .

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                                 chart 1. 《 Group theory color chart version 》2.2 Hold the rectangular card flip game

        example 3.  In the group that describes the symmetry of the magic cube , Altogether 6 There are three basic functions , Each face rotates clockwise 90 degree . Based on this 6 A combination of basic functions （ That is, the binary operation between group elements ） It can generate all possible operations for the magic cube .

2. The drawing of Kelley diagram

         Based on the generator , The drawing steps of Kelley diagram are as follows ：

1. Select any element as the starting point . in fact , No loss of generality , Select the unit element of the group as the starting point . however , Choose any element as the starting point to draw the Kelley diagram, which is the same .
2. Start from the existing nodes , After interacting with each generator, we get a new element , Or go back to the existing element . Use arrows of different colors to represent the generator . If it is a reversible generator （ That is, the inverse of the generator is itself ） Use two-way arrows to indicate , Otherwise, it is indicated by a one-way arrow .
3. Repeat step 2, Until no new nodes are generated （ For finite groups, this step must be reached ）. That is , All the group elements have been described .

         example 4. With cyclic group ​​​​​​​. First draw the representation 0 The starting node of the element ; And then from 0 Process and 1 The role of 1（ Because there is only one generator , So from 0 There is only one arrow for departure . The following are the same as ）; And then from 1 Process and 1 The role of 2;...; Last , from 5 Process and 1 The role of 0. So far, all group elements have appeared , We got Carletto , As shown below ：

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                                                               chart 2  Carletto

          example 5. You can also get the drawing 1 The group of symmetries of the game shown in the Keller graph         

        First , Take the state of cards as the node to draw the Kelley diagram

chart 3 Kelleys diagram of a group that describes the symmetry of a rectangular card flip game （ Take the card status as the node ）

          further , Abstract the above figure from the point of view of function . With B Indicates a horizontal flip （flip horizontally）, With R Indicates a vertical flip （flip vertically）. Take the node in the upper left corner as the unit element , The remaining nodes represent the combined action results obtained from the unit element through the action path shown in the figure . From this, we can get the following figure （ In the figure N Unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit unit ）：

  chart 4  Kelleys diagram of a group that describes the symmetry of a rectangular card flip game （ Take the action operation result as the node ）

        We can know from the Kelley diagram ,BR=RB. Of course, we have until this group and Klein Quaternion group （V4 or K4） It's isomorphic .

 

          notes ： No matter how hard you try to describe the effect, you can't achieve 《 Group theory color chart version 》 One percent of the wonderful degree of the description in ^-^. Interested friends are recommended to read 《 Group theory color chart version 》 original text , Although it takes some time and patience .

   

3. Does the group have the concept of dimension ？

        In linear algebra , A basic attribute of vector space is its dimension , The dimension is equal to the number of basis vectors in the basis of the vector space . As mentioned above , All elements of a group can be generated only by the combination operation between generators . therefore , The generator of a group is similar to the base vector of a vector space . The natural problem is , Is there a concept of group similar to the dimension of vector space ？

        The answer is no .

        Why? ？ Because the size of the set of generators of a group is not an invariant . The size of the basis of vector space is an invariant , That is, a vector space can be composed of different bases , However, different tensors must have the same size of the basis of the vector space . And the group does not meet this condition . With , As mentioned earlier ,{1} Is a set of generators . however ,{2,3} It is also a generator set that can generate it , As shown in the figure below ：

     chart 5  Generate In several different ways （《 Group theory color chart version 》 chart 6.5）

         It can be generated by a generator , It can also be generated by two generators . The size of the set of different generators of a group is not necessarily the same , Therefore, groups have no concept similar to the dimension of vector space ！

4. Track and cycle diagram

        In this section, we'll talk about other things besides kelletto , Kelley Watch （ Multiplication table ） Outside the first 3 A visualization technology ： Cycle chart .

4.1 orbital

        Starting from the unit element node of Kelley graph , repeat

4.2 Cycle chart

        （ To be continued ）

Go back to the main directory of this series ： The foundation of group theory is quickly completed (A crash course for group theory)（1）https://blog.csdn.net/chenxy_bwave/article/details/122702319