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In an undergraduate Abstract Algebra class, it appears that there are two standard approaches: "groups first" or "rings first". Most abstract algebra texts with a "groups first" approach start with some combination of groups in $\mathbb{C}$, groups from modular arithmetic, groups from permutations (including dihedral groups), matrix groups, and then generalize to abstract groups.

Question 1: Why not free groups with relations from the beginning?

Rationale: Working with a free group is almost as general as working with an abstract group, except that you get to explicitly state, for instance, that nothing is commutative, or what the order of an element is, etc. Like $S_n$, all groups can be constructed out of free groups; possibly in a way that's more intuitive than the $S_n$ construction. The popular focus on permutations is great when thinking about groups as acting on something, but it seems that free groups capture the more basic idea that groups are sets that have additional structure and that certain strings of elements are (or are not) equal to other strings of elements. Granted, this is very algebraic/symbolic. But compared to the standard geometric/permutation approach, it seems understanding the awesome generality of free groups would better aid in understanding the properties of a general group.

Question 2: So, does anyone know of an undergraduate abstract algebra text that starts with free groups, relations, presentations, etc?

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    $\begingroup$ Thoughts on Q1: Humans are better at generalising from examples than at making general principles concrete. Five very diverse examples of groups (preferably with a surprise isomophism) leading to a set of axioms ought to be more helpful than being able to manipulate algebraically first. $\endgroup$
    – AndrewC
    May 1, 2014 at 15:56
  • $\begingroup$ I do not know of such a text, but it sounds marvelous. Some humans do better to have the abstract definition first followed by examples so that upon a second reading it settles more deeply. $\endgroup$
    – James S. Cook
    May 3, 2014 at 19:11
  • $\begingroup$ This seems a bit like asking "Why do we teach number theory as being about a model of the Peano axioms, rather than about being the theory of the smallest set containing the empty set and closed under $x \mapsto \{x, \{x\}\}$?" (That's a bit unfair, because I agree that the generators-and-relations approach has uses beyond foundations.) As Paul Garrett says below, when one meets a group in nature, it is much more likely to be because it permutes something than because it is generated by elements satisfying certain relations. $\endgroup$
    – LSpice
    Aug 29, 2014 at 22:42
  • $\begingroup$ Oops, of course I meant $x \mapsto x \cup \{x\}$, not $x \mapsto \{x, \{x\}\}$. Anyway, I think (no surprise!) a better analogy would be Tao's observation (terrytao.wordpress.com/2010/01/01/…) that "At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate." $\endgroup$
    – LSpice
    Aug 30, 2014 at 3:26

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Describing groups by generators and relations (i.e., as quotients of free groups) is not a good way to describe them most of the time. There are some exceptions, Coxeter groups, braid groups, but mostly groups do not arise in that fashion. Further, "the word problem" is undecideable, so it's not the case that there are algorithms (much less efficient ones) to tell whether different presentations give the same group.

The applications of group theory to the rest of mathematics are mostly not about the combinatorics of strings, which is what generators-and-relations defaults to. Invariably, there are structural features of groups-that-we-care-about, which are often not "merely combinatorial", but refer to often-occurring supplemental features.

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I think the answer to Question 1 is that just as numbers are fundamentally about quantities and topology is fundamentally about space, groups are fundamentally about symmetry. We often overlook the "what it's about" part of a branch of mathematics when we look for clean ways to exposit the material, and we can then lose sight of the fact that if we don't know what the math is about then we don't understand it at all.

So, why don't we teach free groups first? For exactly the reason you think we should: "[it] capture[s] the more basic idea that groups are sets that have additional structure." Unfortunately, from this point of view most theorems in group theory seem like they are silly. For example, why all the emphasis on homomorphisms rather than merely functions between groups? If they're really just sets, functions seem fine. From there, why all the emphasis on normal subgroups? Why not arbitrary subsets? Even if you had the idea to look at normal subgroups, why worry about factor groups?

At the end of the day, while groups may not care how you present them, group theory has teleology built into it. From the point of view of the developed theory of groups, a group is, fundamentally, something that codifies symmetry. If our job as instructors is to introduce our students to mathematical culture, then, we can't ignore the teleology of the theory when we present it.

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If you ask, "Why not begin with free groups?" You might first consider what a free group is.

Although you can define them in term of equivalence classes of words, their true definition is in terms of their universal property. I have a hard enough time getting grad students to digest this. Good luck with undergrads first learning what a group is. Related to this, have you considered how difficult it is to prove anything about a group defined by generators and relations?

Dummit and Foote's Abstract Algebra introduces generators and relations informally early on, but even here we don't really ever prove things about these groups rigorously. Issues like how we know $A\not=B$ cannot be addressed without appealing to the universal property.

Plus if you want clean generality, why not just do permutations (i.e. 1800's style)?

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Lots of groups, important for lots of applications, are ugly when presented as generators and relations. Will you teach the key example of:

  • the free abelian group on $\{a_n: n\in\mathbf{N}\}$ modulo the relations $n a_n = a_{n-1}$

If not this, how do you present $\mathbf{Q}$?

How do you get students comfortable with an uncountable set of generators for $\mathbf{R}$?

What generators and relations would you choose for $M_n(\mathbf{C})$?

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I agree that free groups are "cleaner," but what you are looking for here is to get (beginning) students to grasp the subject. This means showing concrete examples, and that the whole theory build up around groups/rings isn't just idle speculation but has down to earth applications. I learned much of algebra from Richman's "Number Theory: An Introduction to Algebra" (Brooks/Cole 1971; sadly long out of print, we got a photocopy of the book as text for the class way back in '85 or so). To see how applying the ring or group ideas simplifies otherwise mysterious, unmotivated proofs is an eye opener (e.g. Euler's theorem is just a one-line corollary to Lagrange's theorem that the order of a subgroup divides the order of the group).

Just my 2¢ as a non-specialist, mind you.

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    $\begingroup$ Did Richman's book give you any constructive insights? Or did he write it before he became explicit about working constructively? $\endgroup$
    – user173
    May 3, 2014 at 21:25
  • $\begingroup$ @MattF, it is an introductory number theory textbook, it doesn't go into such depths. $\endgroup$
    – vonbrand
    May 3, 2014 at 21:54
  • $\begingroup$ Even an introductory textbook has plenty of room to go badly non-constructive (e.g. some proofs of the infinitude of primes), or pleasantly towards constructivity (e.g. in emphasizing the Euclidean algorithm) $\endgroup$
    – user173
    May 3, 2014 at 23:06
  • $\begingroup$ @MattF, I'm not into that particular holy war, but I remember mostly concrete, constructive proofs. $\endgroup$
    – vonbrand
    May 4, 2014 at 2:37
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Question 1: I wouldn't want to present everything in free groups. However, as an early example it could be useful as it takes you outside the world of things which are necessarily finite. Also, if your goal is to expose students to abstract examples, to take them outside their comfort zone, then it's probably a nice idea. So it depends on your audience. What kind of student are you challenging? Can they accept a challenge or do they need to stay closer to the world which they already have experience? The word problem, the problem of deciding if two presentations are equivalent is a bit unsettling to students who want cut and dried answers of how to solve problems. In short, it seems to me a few free groups in your lexicon of examples adds a certain depth to the course which is nice. Especially for those students who go on to work with free groups in later courses. That said, using only free groups introduces needless technical complications that would obfuscate the true geometric origin of many examples.

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