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I am reading about Abelian groups

So apparently it is a set, with an associative binary operation, and identity element, an inverse operation and the binary operation must also be symmetric.

But it is not clear to me how they are useful. Trying to find why they are important it seems they arise as "additive structures" in various systems but this is too abstract for me.

Could someone give some less formal/more practical application or usages showing what is important about abelian groups (layman's terms basically)?

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Informally speaking, an abelian group is a place where you can sum. It's the essence of what defines the sum of ordinary 'counting numbers'.

This allows you - after some exposure - to handle very abstract objects with the same familiarity you have for the latter.

As for any abstract definition, it is not going to make sense unless you have some examples at hand. And as for most abstract definitions, the examples came first. Many, many objects come with a binary operation that is associative and commutative (with unit and inverses) so we gave it at name.

Any "number system" $R$ (technically, I mean a ring here) has a notion of sum $+$, and $(R,+)$ is an abelian group. This includes very familiar number systems such as the integers, rational, real and complex numbers.

But is also includes for example matrices over these number systems. In general, product of matrices is known to depend on the order of the factors, but not their sum. Hence, we can sum matrices 'as if they were numbers'.

Another example is the "number system" $\mathbb{Z}_{12}$, which behaves like hours in a clock. You can think of it as the numbers from $1$ to $12$, but here e.g. $11+4 = 3$. It may be uncomfortable to work with this at first, but knowing that $+$ behaves similarly to ordinary numbers helps.

The list goes on, of course, but it becomes more abstract.

Another thing which may be useful to think about is... well, non-abelian groups. As we said before, in general for matrices $A,B$ we have $AB \neq BA$. Things get non-abelian really quickly in real life too: it is not the same to put your jacket on first and then your t-shirt than doing so in the reverse order.

Commutativity is far from a 'given', hence it is important to know when it does hold. It makes some things easier to organize. But as I said before, if you are not convinced that groups are important in the first place then it may not be clear why them being abelian is a thing to care about.

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    $\begingroup$ The important point is that in an abelian group, a sum is independent of the order of the summands. Your strapline does not capture this point. $\endgroup$ – Rob Arthan Mar 7 at 22:42
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    $\begingroup$ What have rings got to do with this? $\endgroup$ – Rob Arthan Mar 7 at 22:51
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    $\begingroup$ @RobArthan to be clear, I see how my previous comment may indicate that abelian groups are only imporant because they come up studying rings. That's not what I tried to convey. So I've deleted it. But I still think it's a good way to motivate this given the context of the question. $\endgroup$ – guidoar Mar 7 at 22:56
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    $\begingroup$ You are indeed correct (disclaimer: I know little history of mathematics, perhaps someone can give a better answer). In fact, it took a long time before people defined groups the way they are defined now. Iirc originally - as in for example when Galois invented Galois theory - groups were thought as "certain ways to permute/transform variables and operate with solutions of certain equations". Later on the abstract definition of groups was given. Cayley's theorem says that in fact any group is embedded into a group of permutations; hence the two 'definitions' are actually the same. $\endgroup$ – guidoar Mar 7 at 23:44
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    $\begingroup$ (cont.) the Wikipedia article on the history of group theory quotes Galois on using the word group for the first time. $\endgroup$ – guidoar Mar 7 at 23:45
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I'll start with the assumption that you think that the integers $\Bbb{Z}$, the rational numbers $\Bbb{Q}$, and/or the real numbers $\Bbb{R}$ are useful or interesting. All of these are examples of Abelian groups. An Abelian group is just an arithmetic system where "addition" makes sense (and is commutative, associative, etc.). It is a common idea in math to take an object of interest and to study an object one step more abstract. For instance, we formalize in the properties of an Abelian group the fundamental properties of $\Bbb{Z}$. Then, any theorem we can prove for Abelian groups $A$ will apply to any of the examples we care about, like $\Bbb{Z}$.

One might reasonably argue that proving results for $\Bbb{Z}$ on its own might be easier than proving more general results about Abelian groups since more should be true in a specific case. This is true to an extent, but sometimes in stripping down objects to their core properties the important features are no longer hidden by the less important features. This is the sort of thing you start to realize with experience, I think.

More concretely, we know that to study divisibility in the integers, modular arithmetic is essentially indispensable. While we could try to formulate theorems about $\Bbb{Z}$ and $\Bbb{Z}/n\Bbb{Z}$ (integers modulo $n$) separately, we would quickly begin to realize that we are wasting time as there are similar structures common to both of these objects. The notion of Abelian group allows us to treat $\Bbb{Z}$ and all of the $\Bbb{Z}/n\Bbb{Z}$'s on even footing.

Sometimes, it is also more conceptually simple to think of things in a more abstract manner. The polynomial rings $\Bbb{Q}[x],\Bbb{R}[x],\Bbb{C}[x]$ all have a division algorithm which gives them the structures of a Euclidean domain. When considering this, it is impossible to miss the resemblance to the division algorithm we learn in school for $\Bbb{Z}$. Viewing these objects as being members of the same family of objects allows us to prove theorems for them all at the same time and to compare and contrast their individual properties. For instance, we can compare the notions of prime factorization in these rings.

Finally, you might think about this like learning a foreign language in some sense. If you only speak your mother tongue, you will have a strong grasp of it no doubt. However, once you begin to learn other languages you can appreciate the features they have that your own language does not. In turn, this typically gives you a deeper appreciation of the structure of your own language. I would say it's the same thing: learning about Abelian groups will change the way you understand polynomials and the integers and enrich your knowledge of them.

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  • $\begingroup$ What does the notation $\Bbb{Z}/n\Bbb{Z}$ mean exactly? Also how does a ring differ from a polynomial ring? $\endgroup$ – Jim Mar 20 at 11:17
  • $\begingroup$ $\Bbb{Z}/n\Bbb{Z}$ is a notation for the integers modulo $n$. There are many rings that are not in an obvious way polynomial rings. For instance, take $C^\infty(\Bbb{R})$, the smooth functions on the real line. $\endgroup$ – Alekos Robotis Mar 20 at 20:29
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The same question could really be asked about any abstract algebraic structure: why do we study fields or rings or groups or semigroups, etc.? For that matter, why do we study abstract algebra at all?

As other answers have noted, algebraic structures are studied because they are useful abstractions. For example, the reason why we study groups is because the definition of what counts as a group is:

  1. broad enough that a lot of things in mathematics satisfy it, but
  2. specific enough that we can prove quite a few useful theorems using only the group axioms, which means that these theorems automatically apply to everything that satisfies those axioms.

For example, the integers (and rationals and reals and even complex numbers) under addition, positive rational (and real) numbers under multiplication and invertible $n \times n$ matrices under matrix multiplication all satisfy the definition of a group. This means that any theorems that we prove about groups in general apply to all of those systems, as different as they may seem on the surface.

And we study abelian groups because, as useful as groups are, it turns out that there are also quite a few useful theorems that we cannot prove just from the group axioms unless we also assume commutativity. Adding the commutativity axiom thus allows us to prove more useful results — the tradeoff, of course, being that those results won't apply to groups that are not commutative, such as the group of $n \times n$ invertible matrices mentioned above.

This is a common situation in abstract algebra (and in many other areas of "abstract" mathematics): the more specific we make our abstractions, the more results we can prove, but the fewer things those results will apply to. The trick here is to find abstractions that happen to hit a sweet spot where there are just enough axioms to prove useful results, but no unnecessary ones that we could remove and still prove essentially the same results. Of course, in practice, there are many such "sweet spots" of varying specificity. Experience has shown that groups are certainly one of them, while abelian groups are another.

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  • $\begingroup$ While at the begin, I follow your thoughts (very helpful) which is about how abstractions help and how we can generalize I get confused here: For example, the integers (and rationals and reals and even complex numbers) under addition,. If I understand correctly this is a "constraint" right? So we can apply the group axioms etc only if we consider summation no other operation for integers/reals etc? How does that integrate with the specific number system we work on? $\endgroup$ – Jim Mar 20 at 11:22
  • $\begingroup$ Adding the commutativity axiom why is the word "axiom" used here? Isn't the term "requirement" more appropriate? Or am I confused here? $\endgroup$ – Jim Mar 20 at 11:23
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    $\begingroup$ @Jim: The definition of an (abelian) group consists of a list of statements called axioms, which are taken to be true by definition. In particular, we define an (abelian) group to be any system that satisfies those axioms. From these axioms we can then prove various theorems, which we thus know will hold whenever the axioms hold. $\endgroup$ – Ilmari Karonen Mar 20 at 13:40
  • $\begingroup$ … The theorems themselves can be studied as an abstract logical theory that arises from the axioms alone and exists independently of any concrete models that it may have. But in practice, what we often want to do is show that some specific structure (or a class of structures) satisfies the axioms, and thus that all theorems arising from those axioms also hold for this structure, and then use that fact to show something useful about the structure. $\endgroup$ – Ilmari Karonen Mar 20 at 13:42
  • $\begingroup$ Since we define what is a group and what must be true for something to be a group, would it be correct to state that axioms are equivalent to constraints that must hold true? $\endgroup$ – Jim Mar 20 at 16:49
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When considering the notion of Abelian groups, you are dealing with an algebraic abstraction: you will not find in mathematics any structures that are just Abelian groups without being something more specific, just like you won't find any animals that are mammals without belonging to some more specific species. The importance of an abstract notion is not that it describes something that cannot be described in a more specific way, but that the notion captures common properties of distinct structures in a useful way. Useful here means that it can explain certain properties common to all these distinct structures that follows the exact same pattern of explanation, even though the structures may be quite different. The same principle holds for all kinds of abstract notions, like (to stay within algebra) fields, vector spaces, rings, groups and whatnot.

The importance of the notion of Abelian groups then is that for certain purposes it captures a sweet spot: it is sufficiently restrictive to prove properties that do not hold for more general groups, while it is not so restrictive that it excludes numerous examples which, for these specific purposes, behave in a very similar way. It is not immediately obvious that Abelian groups do serve such a purpose, but experience shows that on one hand one very often encounters examples of Abelian groups, and on the other hand very many interesting mathematical questions can be treated by arguments that apply uniformly across all these examples, while they are not applicable to the even broader class of groups in general. Basically all structures where one can do the customary additive bookkeeping, adding and subtracting items at will, without constraints on the possibility to perform these operations and with associativity and commutativity guaranteeing that the order of such operations does not matter, are Abelian groups.

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  • $\begingroup$ Very helpful answer. but experience shows that on one hand one very often encounters examples of Abelian groups, what are some well known examples in various domains i.e. not only mathematical but other principles as well? $\endgroup$ – Jim Mar 20 at 11:29

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