I watched several videos but I still can’t understand the depths of the subject. I know its about groups and rings and other structures. While I do understand it is pure mathematics, it must have some application in other parts of pure mathematics as well, right? How is abstract algebra used in other (math) topics? I don’t understand the true depth of abstract algebra.
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How do you take two things and make another thing in a way that is meaningful in some capacity?
Can you identify patterns/structures in such an operation in order to come to meaningful conclusions?
Can you relate one thing to another based on how the properties of how they combine rather than other superfluous information?
Abstract Algebra deals with these questions.
For example, if you have a square then you can rotate it/flip it and put it back down. This is a symmetry of the square. For example, you can flip it horizontally and vertically. The thing is, you can take two such symmetries, do them in succession, and you’ll get back a *new* one. For example, if you first flip horizontally and then vertically, then you will have actually done a 180 degree rotation. So hFlip + vFlip = 180Rot. Pretty cool. Because of this, the symmetries of a square are objects of study in abstract algebra.
Another thing we can look at are Complex Numbers. Specifically, if we have the numbers {1,-1, i, -i}, then I can multiply these together and always get another one of these things. Eg, -1*i = -i. So these have some kind of structure. These also are objects of study in abstract algebra. In particular, if you track how these work on by plotting these on the complex plane you can notice a few things. Firstly, they all look like some kind of rotation. 1 = 0 degree rotation, i=90 degree rotation, -1 = 180 degree rotation, -i = 270 degree rotation. Cool. But another thing you might notice is that if you draw lines between these on the complex plane, then they make a square and so their multiplication produces symmetries of a square! This means that these things and the symmetries of a square are actually related to each other even though they come from totally different concepts. If I can understand each of these separately and the specifics on how they relate then I might be able to arrive at nontrivial knowledge about both! This is, in general, the objective of abstract algebra.
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