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In the most basic sense, what is abstract algebra about?

Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields."

I find this, however, to say the least, not very informative. What do they mean by abstract algebraic structures? Along these lines, what are groups, rings, and fields then?

I've been told by a friend that groups, essentially, are sets of objects, although, this still leaves me wondering what he means by objects (explicitly).

I don't need anything rigorous. Just some intuitive definitions to give me some direction.


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Sets plus some additional "structure" that lets the elements be combined or related –  Hamton Jul 3 '14 at 22:03
@Pilot Would we say it's an "ordered" set then? –  Colin Jul 3 '14 at 22:31
@Colin An ordered set can refer to one of several related concepts. For what you're probably thinking about, there is a totally ordered set which is a set equipped with a binary relation $\le$ such that three axioms hold: $a \le b$ and $b \le a$ implies $a = b$; $a \le b$ and $b \le c$ implies $a \le c$; either $a\le b$ or $b \le a$. The reals form a totally ordered set, for instance. –  Arkamis Jul 3 '14 at 22:58
Yeah you could say you are studying the "order"/ "organization" of a set in abstract algebra, in a very general way. Order usually is more specific though, see Arkamis's comment –  Hamton Jul 3 '14 at 23:00

6 Answers 6

up vote 27 down vote accepted

We learn math with numbers early on. We learn how to apply operations to numbers to get new numbers. We learn rules, and consequences of those rules. All of that is pretty straightforward.

But, the real numbers are not the only things we might want to examine in detail. The properties of how elements interact under operations is a more general, abstract notion of what we do with numbers when we do algebra.

For instance, maybe we want to examine what a shape looks like if we rotate it around. Maybe you run a supply chain, and you need to build 4 widgets, but only some of those widgets need to be built in a certain order. Could you re-arrange things to make it more efficient? Maybe we want to explore structures that have a fundamental periodicity, like the time of day.

Over time, we have constructed concepts of structures that elements can belong to, and notions of operations on these structures. These structures -- groups, fields, rings, monoids, modules, vector spaces, etc. -- don't have a natural set of rules, per se. We make up those rules (aka axioms), but we have found that many natural concepts adhere to those rules.

This is all well and good but somewhat useless until you learn about isomorphism. Exploring what a group is or what a ring is is fine. But the richness of abstract algebra comes from the idea that you can use abstractions of a concept that are easy to understand to explain more complex behavior! Adding hours on a clock is like working in a cyclic group, for instance. Or manufacturing processes might be shown to be isomorphic to products of permutations of a finite group.

Abstract algebra is what happens when we want to explore consequences of rules and properties on collections of objects of any type -- hence the term "abstract!"

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In "concrete" algebra one works with things like integers, rational numbers, real numbers, complex numbers, matrices, quaternions, permutations, polynomials, geometric transformations (e.g. isometries, similarities, reflections, inversions, projectivities, etc.), etc., subject to operations like addition, multiplication, and composition.

In "abstract" algebra one says "suppose we have a set of objects (which could be numbers, matrices, permutations, geometric transformations, etc., but we will not say what they are) and certain operations (which could be addition, multiplication, composition, etc., but again in certain contexts we won't say what they are) that are assumed subject to certain algebraic rules, such as commutativity, associativity, distributivity, the existence or non-existence of identity elements and inverses, closure or lack of closure, etc.). Then one deduces consequences of those algebraic laws. Statements that say that something is always true are deduced from algebraic laws without considering the concrete nature of either the objects or the operations. Statements that say that something is not always true are often deduced from concrete examples, involving numbers, matrices, polynomials, permutations, etc.

In abstract algebra, the examples are concrete, but the derivations of general results come from the rules of algebra without the concrete nature of the operations or the things they operate on.

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Mathematics has to do with sets. There is no definition about what the set is, but we all know that set is made up by elements. Many problems in nature can be represented by sets, and the relations of the elements on these sets. The mathematical discipline that studies THE RELATIONS OF ELEMENTS on a given set is called algebra. There are many good properties of relations that a set can have, and from those properties we classify classes of 'sets with their relations' (algebraic structures) and call them as groups, rings ect.

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When you first came to learn maths, what did it show you? For instance, we wrote a lot of equations in terms of x,y,z and so on, but essentially, what they had common is that they were representing some unknown numbers.

Abstract algebra is a bit more broad concept: here (in intuitive sense) letters represent pretty much anything, rather than numbers. I guess you might refer this 'everything' as 'objects'. So why abstract? What all abstract algebra has in common is that they study some fundamental properties of set of objects. We discard anything that is not relevant: for instance, in study of groups we only concentrate on properties of groups; no distributive laws or definitions of addition and multiplication. We only need a binary operation on a set and associativity, inverse and identity.

By discarding any irrelevant features of objects, we can concentrate on essential properties of them, and anything derived from these essential features will be applied to vast amount of things that have these elements in common. This is the power of abstract algebra, however by doing this we inevitably deal with more abstract things: hence abstract.

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The other posters did a good job explaining what abstract algebra is, so I'll try to help you understand groups. You can think of a group, ring, field etc. as being a set with a certain structure attached to it. The intuition usually comes from concrete examples so I'll include a few. (Disclaimer: This will not be rigorous in the slightest. I'm shooting for conceptual clarity.)

A group is a set together with a binary operation that we often think of as multiplication (or addition in some cases), satisfying certain properties:

Closure: if you multiply two elements of the set, you get another element of the set. For example, the set of positive real numbers together with the operation of ordinary multiplication forms a group. This group is closed, because if you multiply two positive numbers, you get another positive number.

Identity: There is an element in the set called the identity that works like 1 (or 0 in an additive group). That is, if you multiply (add) it by an element you get the same element back.

Inverses: You have to have a way of getting back to where you came from. If you're in the group of positive real numbers, the inverse is just the reciprocal. e.g. 1/4 is the inverse of 4. Here's another example. The set of rotations of the plane, together with the operation defined by: rotation2 * rotation1 = (do rotation 1 first, then do rotation 2), is a group. If I rotate the plane 90 degrees counterclockwise, the inverse is just a 90 degree clockwise rotation, which gives you back the identity.

Associativity: This one might seem a bit obvious, but it has very important consequences. If you have elements a,b,c in your group, then a*(b*c)=(a*b)*c. i.e if you multiply b*c by a, you get the same thing as you'd get multiplying c by a*b.

Source: A Book of Abstract Algebra, by Charles C. Pinter. This is a great intro book, if you want to learn more.

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Pinter in "A book of abstract algebra" says it thus:

Thus, we are led to the modern notion of algebraic structure. An algebraic structure is understood to be an arbitrary set, with one or more operations defined on it. And algebra, then, is defined to be the study of algebraic structures. It is important that we be awakened to the full generality of the notion of algebraic structure. We must make an effort to discard all our preconceived notions of what an algebra is, and look at this new notion of algebraic structure in its naked simplicity. Any set, with a rule (or rules) for combining its elements, is already an algebraic structure. There does not need to be any connection with known mathematics. For example, consider the set of all colors (pure colors as well as color combinations), and the operation of mixing any two colors to produce a new color. This may be conceived as an algebraic structure. It obeys certain rules, such as the commutative law (mixing red and blue is the same as mixing blue and red). In a similar vein, consider the set of all musical sounds with the operation of combining any two sounds to produce a new (harmonious or disharmonious) combination.

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This is exactly the excerpt I had in mind when I read this question. Pinter does a wonderful job of motivating the subject. –  Ryan L Jul 10 '14 at 17:09

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