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I am preparing to give an abstract algebra course, which should be mainly focused on Group Theory. However, I want to cover other algebraic structures (magmas, quasigroups, etc.) as well.

I am wondering when I should cover them. If I do it in the beginning, it will connect nicely to the other basic properties of a group other algebraic structures may or may not have. However, I am afraid that an overload of algebraic structures will confuse the students. But I have to cover monoids somewhere in the beginning anyway to get to the free monoids, which are usually covered somewhere in the beginning. On the other hand, if I do it in the end, then there will be some repetition of the basic properties, which is nice.

How should I approach this?

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    $\begingroup$ If it were me, I'd talk about rings, fields, vector spaces, modules, and such things (because the students will already have an idea about many natural examples), rather than playing the game of weakening axiom systems and naming things, since I feel that the latter gives many students the wrong impression about how mathematics works. $\endgroup$
    – paul garrett
    Dec 21, 2015 at 19:17
  • $\begingroup$ Is this an undergraduate-level course, or a graduate-level course? $\endgroup$
    – mweiss
    Dec 21, 2015 at 19:26
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    $\begingroup$ @mweiss Undergraduate, second year of university. $\endgroup$
    – wythagoras
    Dec 21, 2015 at 19:31
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    $\begingroup$ I asked only because you mentioned that free monoids are usually covered in the course, which (in my admittedly limited experience) is not typical for a first semester undergraduate course in abstract algebra. $\endgroup$
    – mweiss
    Dec 21, 2015 at 19:33

2 Answers 2

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I think you are correct to worry about an overload of algebraic structures, especially if they are not well-motivated. I would strongly encourage you to keep the naming of various algebraic structures to a minimum in a first course in abstract algebra. In my view, the goal of the course is for them to see the power of abstraction in a fairly concrete setting, illustrating how it illuminates certain structures (e.g. the study of symmetry via group theory, the similarity in structure between integers and polynomials in one variable with coefficients in a field via ring theory). I think it is much more useful to get to something non-trivial and engaging for the students such as the wallpaper groups, Platonic solids, or Rubik's Cube rather than trying to convince of the austere beauty of abstract algebraic structure for its own sake.

If you insist on introducing a bunch of different structures, my guideline for when it is appropriate to move on from one structure to another is when students, at a minimum, can do the following: (1) distinguish between objects which have the structure and which do not. (e.g. is such-and-such set with such-and-such operation a quasigroup? is such-and-such subset of a group a subgroup?) (2) be able to work with simple ideas related to the given structure such as substructures, quotient structures (though this is challenging in a first course) and homomorphisms. (e.g. is the following set-theoretic map a homomorphism of groups? is the following quasigroup a Moufang loop? what is the kernel of the following group homomorphism?)

If students do not become proficient at both (1) and (2) for a given algebraic structure, then it is not clear to me what is the point of introducing said algebraic structure in the course. As I suggested in my first paragraph, I am very skeptical that students can actually acquire significant mastery of multiple algebraic structures during the course of a semester. I think it is much more valuable to students to understand group theory at a deep level than to have a broad, but shallow, acquaintance with a wide variety of algebraic structures.

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When I took a first course in abstract algebra, we were taught the words "semigroup" and "monoid" along the way to learning what a group is. This was mainly a matter of convenience of language. It allowed us to say things like, "A ring is an abelian group under addition and a semigroup under multiplication". (Our rings didn't need an identity.)

The actual theory of semigroups is of course a giant topic, almost all of which I think is not appropriate for a first group theory course.

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  • $\begingroup$ If they are introduced just as a shorthand of an aspect of rings, better leave it out. Just adds to confusion (extra terminology with little or no justification). $\endgroup$
    – vonbrand
    Dec 22, 2015 at 18:00

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