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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> it must have some application in other parts of pure mathematics as well, right? How is abstract algebra used in other (math) topics?
The main value of abstract algebra is that it allows you to generalize and prove results about a whole set of structures instead of just one specific structure. For example, there are many examples of groups throughout mathematics. Rotations are a group. Invertible matrices of a given size are a group under the operation of multiplication. Complex numbers are a group under addition, and nonzero complex numbers are a group under multiplication. And so on, and so on. Anything we know about groups in general automatically applies to all of these specific groups. For example, we can show from the definition of a group that it cannot have multiple identity elements, and that no element can have multiple inverse elements. So this result automatically copies across to all of those different groups I mentioned.
It can also make proofs simpler and help explain how they work, since you’re only assuming the minimum set of properties needed for the proof. For example, if I asked you to prove from scratch that a specific 15×15 matrix has a unique inverse, you might not know where to start. But the proof that any element of any group has a unique inverse is trivial. It’s one of those “can’t see the forest for the trees” situations. Throwing away some of the details of the structure we’re working with and focusing on a few specific properties can make some aspects of the structure easier to understand, especially if people have already spent ages studying those properties in a different context.
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