How can I visualise groups in Group Theory

I'm having a hard time grasping groups in Group Theory. Is it ok to visualise them as being sets with the group axioms and a binary operation, intuitively as a venn diagram? and G without * is really confusing me. I can't seem to move on in my study of Group Theory because of this. Can anyone also recommend me any online sources for a clear understanding on groups please. What are some applications of groups? how can I apply them and see them to better understand the purpose of them. I feel really anxious whenever I open my textbook on Group Theory and it's because of these answers that I'm missing. I've looked through countless books and online notes. Everything is seems too complex and doesn't sit in my mind. I would love to study this subject more effectively. I had this problem while studying Set Theory and I need another approach for Group Theory. Thank you!

Solution 1:

Groups are symmetries. The best way to think about a group is that you have an object that has some kind of symmetry, and the group represents mathematically that symmetry. For example, $D_8$ is the symmetry of a square, with each element representing a rotation of flip of the square; $C_5$ is the symmetry of five objects rotating in a circle, with each element representing a different rotation; and $S_7$ is the symmetry of $7$ points that can be permuted in any fashion, with each element corresponding to a different reording.

In my opinion, this is far and away the best way to think about groups, even abstractly. You have some unspecified object and each element of the group represents a way that an object or collection of objects can be spun, flipped, reflected, etc. without changing.

See here for a cute interactive illustration of $D_8$ being the symmetry group of a square.

Solution 2:

The video "Visualizing Group Theory 1" is in my opinion an almost ideal visualization-and a good introductory summary-of groups.

Also, reading the classic book "Symmetry" by Herman Weyl is a must.

Solution 3:

I am almost sure that group theory was developed to describe transformations of symmetric objects.

For example let's take a cube. Let's consider all the rotations of the cube such that the cube stays in place. F.e. you can rotate it around vertical axes $z$ by $\pi/2$. Or you can rotate it around horizontal axes $x$. What if you combine these two transformations? You would get some new rotation, probably around some diagonal of the cube.

You can rotate the cube around the $z$ by $5/2\pi$. After that it's position would be the same as after rotation around $z$ by $\pi/2$. Let's count such rotations identical because they produce the same final position.

How many different rotations of cube are possible? (hint: 24)

1. Let's say that if we did not rotate a cube at all this IS a rotation ("identical" rotation).

Note that:

2. For every two rotations combination of these rotations is also rotation.

3. For each rotation there is an 'opposite' rotation, which returns the cube in original position.

Note, that these three properties resemble very much the definition of group.

Cube is symmetric. But there are many symmetric objects, symmetric in their own way, not like a cube. The group of transformation of an object (set of possible transformations and a rule, which gives a result of combination of two transformations) is a mathematical description of "how exactly this object is symmetric".

Another example. Consider Rubik's Cube. There is a group of transformations of Rubik's Cube. Elements of the group are combinations of simple sides rotations (f.e. rotate facade clockwise, then left side clockwise, then back side two times). You can easily check that these transformations form a group.

And as soon as I understand it's a group, I can prove that if you repeat the same transformation many times, sooner or later you will get into original state! (well, I used one more fact: there number of such transformations if finite).

By the way, what is the number of possible states of Rubik's Cube? Google knows, but I can easily prove that it's divisible by 4, because there is a subgroup with 4 elements.

Group theory is cool.

Solution 4:

Groups are just permutation groups. By Cayley's theorem, every group $G$ is isomorphic to some subgroup of $S_n$, the group of all permutations on $n$ elements, for some $n$. The proof is to to associate each $g\in G$ with the function $f_g:G\to G$ defined by $f_g(x)=gx$. The group axioms guarantee that this is a bijection, and the associativity axiom in particular says that $f_{gh}=f_g\circ f_h$.

Thus, if you like, group theory is just the theory of permutations (sometimes on infinite sets). Just swapping eggs around in an egg carton.

Solution 5:

Here is a very good and informative video series about groups: Visual Group Theory.

The series covers a lot of stuff. Everything from the axioms to Galois Theory. Everything is visualized by diagrams and intuitive observations.

Then there is this video. It's the best video I've ever found about applying group theory.

P.s. I know the feeling you have. When I was learning group theory, I was also struggling with the lack of intuition behind the theory. But once you get over it, you'll find group theory super interesting topic.