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.
In: 7
[Modular arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic), for example, relies heavily on group theory and ring theory. Meanwhile, one of its most common applications nowadays is asymmetric encryption (such as RSA).
A good way to think about it is taking concepts we know from math and trying to understand what they mean “in the abstract”.
For example, let’s take the concept of adding integers (positive and negative whole numbers like 0, 2, 7, -25, but not decimals or fractions).
You’ve got all of the integers, and you’ve got this concept of addition. If you add two integers you get a third integer. If you add zero, you get the same integer back. And every number has its inverse – like 7 and -7…when you add them together you get zero.
What if we took those concepts out of the integers?
We’ve got a bunch of “things”. You can perform some operation on two of those things to get a third thing. There’s one “thing” called the identity – if you apply the operation to it and another thing you always get the SAME thing back. And every thing has an inverse.
Is it possible to have those same properties hold for something other than integers?
The answer is yes, there are lots of sets of “things” and operations that satisfy those criteria.
Mathematicians call that a “group”. No particular reason, it’s just the name given to it. And there are “groups” of things that aren’t very much like numbers at all, but still have those properties.
Similar things happen if you try to study addition, subtraction, multiplication, and division and abstract those – mathematicians call that a “field”. It turns out you can define fields over sets of things that look nothing like numbers.
Studying groups and fields (and others, like “rings”) has led to all sorts of interesting discoveries. Some of those discoveries end up applying to numbers – but some don’t. We’ve discovered that many objects in the universe behave like groups, rings, or fields, so much of abstract algebra has turned out to be useful in surprising ways. But other parts of abstract algebra have turned out to be just theoretically interesting….so far.
But that’s the idea. Take concepts we know in math, and “abstract” them, to study their properties in the abstract without necessarily making assumptions about numbers.
> 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.
Imagine you’ve done a lot of work, and proved a lot of theorems about **integers** number, and now someone says “well, integers are too restricted, can I use the same theorem on real numbers?”. Then, you have to go back through all your theorem to prove that they work on **real** numbers.
Latter, someone else says “I love your theorems, but I actually use complex numbers in my work, are your theorems still true with complex numbers?”. So you’re back at your black board and for the third time, you’re going through all your theorems for **complex** numbers.
Then, at a conference, you meet with a theoretical computer scientist, and after some talking, they say to you: “Those theorems would greatly help me, but there is just a problem, we’re not dealing with actual numbers, we’re dealing with those complex data structures. But in the idea, they work just like number, you can still add them, multiply them even if that’s a little weird to do. Do you know if your theorem would also work on those things that are almost numbers?”. So for a fourth time, you’re going through all your theorems and checking that they work for those **almost-numbers**.
And that’s a lot of useless work, because you’ve done **the exact same proof 4 times**, and peoples will keep coming up with slight variations like that are more general, or sometimes less general “What if negative numbers are not possible? Can we still use your theorem if negative numbers are forbidden?”
In my experience, the main point of Abstract Algebra is to try to make the proof once and for all. Instead of saying “I prove my theorem for integers”, you’re saying “I prove my theorem for everything that has an addition and a multiplication, and for that I need those very simple properties like n+0 = n and n*0 = 0, but outside of those simple properties you can do whatever you want. I don’t care if those are numbers, colours, data-structures, or even the weird quantum superposition of numbers that physicists keep talking about. It always works.”.
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.
It’s good for a lot of things. One of the fun applications is code breaking. But yeah it’s not as useful as linear algebra which can be applied in many many places.
Latest Answers