# Intuitive Explanation of what Abstract Algebra is all about [duplicate]

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I looked up wikipedia but honestly I could not make much sense of what I will basically study in Abstract Algebra or what it is all about .

I also looked up a question here : What is Abstract Algebra essentially?

But there are so many definitions and terms that I always get bogged down by them.

It would be helpful to me and maybe others if someone could explain what Abstract Algebra is all about in simple words that one can understand intuitively.

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• Were there any answers to that question that seemed helpful and which you'd like to understand, but which included words whose meanings weren't clear? – user49640 Jun 20 '17 at 9:02
• If by intuitive you mean concrete, then that's inherently difficult: abstract and concrete are antonyms :) – MathematicsStudent1122 Jun 20 '17 at 9:05
• It's funny, because some of these answers talk about "why we chose the number system we chose" or things like that, as if commutativity and associativity were the root reasons for why we have natural numbers. I think the reason we actually count with natural numbers is that they represent (some) equivalence classes for the equipotence relation. (A set $A$ is equipotent to a set $B$ if there exists a bijection between them.) Addition is then defined by associating to $A$ and $B$ the set $A \cup B$ (when the sets are disjoint). The algebraic properties of addition are thus a consequence of... – user49640 Jun 20 '17 at 10:19
• the algebraic properties of set union, which result from logic. In other words, $3 + 5 = 8$ because if you put together $3$ objects and $5$ objects you get $8$ objects. Commutativity, associativity, etc., are a consequence of this, not the reason we thought using natural numbers would be nice. – user49640 Jun 20 '17 at 10:21
• See also this answer. – Bill Dubuque Jun 20 '17 at 14:49

## 5 Answers

Abstract algebra is the study of operations.

As humans, we agree that $a + b = b + a$ makes intuitive sense. We take this rule as an axiom, building our human number system with this axiom as part of the foundation. But maybe aliens from Planet Zog believe that $a + b$ should not be $b + a$, which leads them to create/discover an entirely different alien number system.

Questions arise immediately. Can one really build a meaningful number system if one abandons $a + b = b +a$? What about the other axioms, like $a + (b+c) = (a+b) + c?$ What happens if we get rid of those?

On the flip side: Is there anything we can say about all number systems where $a + b$ does equal $b+a$? More interesting still: How many axioms do we need to take before we are led, inevitably, to the number systems we know and love (like the integers)? The idea that we can describe the integers with axioms alone is very attractive: it would mean, in a sense, that the integers are really god-given.

Abstract algebra addresses questions like these.

• Is abstract Algebra only about about how we invented numbers, and why we invented them that way , or are there many other things in Abstract Algebra and this study is only a part of it ? – Freelancer Jun 20 '17 at 10:05
• Abstract algebra is about the properties (like $a + b = b+a$) of our number system (like the real number system), and the properties of other number systems (that are more abstract). The first question is: If we have some list of properties that we like (such as $a + (b + c) = (a+b) + c$ and $a + 0 = a$), what can we say about all number systems with those properties? That is where abstract algebra starts. – Jesse Madnick Jun 20 '17 at 10:09
• So, it's not really about "how we invented numbers." It's about the properties that our number system has, and about other (strange) number systems that share similar properties. – Jesse Madnick Jun 20 '17 at 10:10
• Actually what I wanted to ask in comment above here is that I get the above mentioned things about operations , number systems etc .. And they seem very interesting also , but what I want to ask further is that is Abstract Algebra the whole subject , only about the above mentioned things or are there many more things in Abstract Algebra other than the study of operations of number systems . – Freelancer Jun 20 '17 at 10:23
• I should really emphasize: The "number systems" of abstract algebra are not always literally systems of numbers. The more honest term would be "algebraic structures," which is a broad term. For example, the set of all rotations of the plane (like, "Rotate by 30 deg" and "Rotate by 45 deg", etc) is itself an algebraic structure. Here, the "operation" in question is just: Do one rotation, then do the second rotation. The point is that the same tools used to study number systems apply here, too. This is the power of abstract thinking. But this thread is long; I should sleep. – Jesse Madnick Jun 20 '17 at 10:39

This is a fairly tricky question as Abstract Algebra is one of those things that makes a lot more sense one you've spent some time studying some of it's subject areas.

I will however have a stab at it...

Abstract algebra normally follows the same pattern of taking a set, $Z$ say, and attributing some properties to the elements of those sets. We then seek to prove certain things about those sets.

We do in fact use sets of these sorts all the time, since number systems are formed in exactly this way. For example, the set of integers $\mathbb{Z}$ is simply a set of numbers with the properties of the numbers being that they can be added and subtracted from each other, with an identity element (an element such that adding it to other elements just gives you what you started with) zero etc.

The goal of the 'abstract' part of abstract algebra is due to the fact that rather than just studying very specific cases, we instead look for common properties across many types of sets and mathematical objects and study the properties that they all have in common. This is similar to a the fact that while there are many hundreds of breeds of dogs, they all have very similar physiologies and so a vet can study the common aspects of all dogs and in doing so she can then perform operations on any of them, even though there are other aspects in which they all differ.

So this why in abstract algebra we don't specifically study each individual set such as $\mathbb{C}$ and$\mathbb{R}$, but instead we notice that they all satisfy certain properties in common. We then define a Field (as one example) as being a set $\mathbb{F}$ with these properties in common (you'll study Fields when you start your course). Then anything we prove about Fields can be applied to all of the sets which also share the properties of a Field.

There are many other types of objects such as as Groups and Rings which are also studied because they have properties that we know exist for many specific cases.

Therefore, in conclusion Abstract ALgebra is the process of noticing similar properties in different mathematical objects and then specifically seeing what we can prove about objects with these properties in order to be able to say things about every object with those given properties.

• This is the answer which I feared I would get "Abstract Algebra is one of of those things that makes a lot more sense one you've spent some time studying some of it's subject areas" ..... To be Honest I really couldn't get the answer which I was looking for ... But I guess I will have to study about it a bit first to make more sense of this – Freelancer Jun 20 '17 at 9:45

Abstract algebra "generalizes" the notion of numbers.

For example, we can add numbers. But we can also add apples. Also, the order of adding numbers does not matter. Similarly with adding apples. We can add zero to any number and nothing changes. We can also add zero apples and nothing changes.

In other words, there are other things in the world that satisfy the same rules as numbers. That makes one think that maybe numbers are just a specific case of some bigger "class" that satisfies certain properties (such as, for example, that we can add two elements in the set). That is what linear algebra is trying to understand; namely, it is trying to find and understand certain characteristic "properties" that somehow distinguish one class of objects from the other.

If I had to explain it to my 6yo daughter, I would tell her that "Abstract Algebra" is about how we invented numbers, and why we invented them that way.

She probably wouldn't understand and tell me to let her play.

• As far as "how we invented numbers and why we invented them that way" ... I used to think it was a philosophical question ? – Freelancer Jun 20 '17 at 9:53
• I wouldn't consider that Philosophy, Epistemology and Abstract Algebra are disjoint fields... – Evargalo Jun 20 '17 at 10:02

You take some kind of set and you define some operations on it so that they have some nice properties that you find interesting . Then you try to learn what happens. Operations, rules and sets are often quite general. You can apply what you've found at things that are not numbers, like set of permutations or rotations, .... It's a pretty cool field but I think that is impossible understand what it is about until you don't study it and I'm pretty terrible at explaining it. This video is funny and I think can give some ideas on group theory, that is a part of abstract algebra https://youtu.be/FW2Hvs5WaRY