Quotients & Homomorphisms for Beginners: Part 1, Motivation

In a typical first course in “abstract” or “modern” algebra (known simply as “algebra” to mathematicians), we introduce students to quotient groups and homomorphisms. In fact, the quotient group is the first group one comes across that is somewhat complicated; most other groups discussed up to that point will be abelian, cyclic, et cetera, and most importantly, students will be at least somewhat familiar and comfortable with the groups beforehand. Therefore, the quotient group represents a unique challenge, a first barrier for students with respect to group theory.

I believe many educators and students would agree with my stance that the quotient group is rarely well motivated, and even more rarely well understood by students, at least while they are in the course (for many, it will ‘click’ later on). This is unfortunate for many reasons. First, the aforementioned challenge is a massive intellectual achievement for those who do grasp the structure. Second, the quotient group, when thought of as an operation of sorts of some normal subgroup N or its parent group G, preserves several important properties of groups such as commutativity and solvability. Third, the quotient group comes in many forms (for this reason, one might prefer to use the plural terminology, but this is inconsequential). Fourth and finally, the overcoming of the challenge of understanding the basics of the quotient group can snowball and proceed to enlighten students of mathematics further by providing a gateway to comprehension of homomorphisms. With few possible exceptions, exempli gratia the fact that all vectors spaces over a field F with dimension n are isomorphic, the so-called Isomorphism Theorems, which primarily concern themselves with quotient groups and related structures, are the first isomorphisms or even homomorphisms of serious substance a student comes across.

At least with respect to group theory and introductory algebra, I feel a true, comprehensive understanding of quotient groups and homomorphisms can provide a foundation of knowledge and a confidence that can propel a student to further success. In mathematics, perchance algebra especially, one often feels (s)he understands something and is supremely confident in that fact, when, in fact, the material is far deeper than (s)he realizes. The more abstract the content and the simpler the beginnings, the worse the predicament. Therefore, courses in general algebra, category theory, and real analysis provide excellent examples of courses where it is quite easy for students to fall victim to their own hubris and fall behind. As such, many students fail these courses. I believe the status quo described hitherto is the primary reason for this, not the difficulty innate to the subject matter, the abstract nature (at least directly), or other commonly cited attributes of courses such as those previously listed.

To ensure I treat this subject with the care I feel it deserves, I have split the discussion into parts. While I lack the foresight to state now how many parts there will be, I would expect it to be around 3. This will come in eight parts. This series is now concluded and consists of six parts. Thus far, I have yet to have any “real math”, that is no equations, no formulas, no theorems, no lemmas, no corollaries, and so forth. This is somewhat intentional. Those who will benefit most from these posts are students studying mathematics at the undergraduate level, and for many of these students, mathematical maturity and experience is sparse. The idea that mathematics should consist wholly of symbolic manipulations or the theorem-proof-theorem-proof format so ubiquitous in some materials is a misconception worthy of breaking. Terence Tao has an excellent blog post that expounds upon this matter, discussing his idea of how mathematics is neither about loose arguments nor entirely rigorous proof, the likes of which even computers could understand. But, I digress. My point is: this post is mathematical despite its lack of those features considered by many to be essential for anything of mathematical substance. Moreover, mathematics a human endeavor, involving language as well as abstract representations of ideas.

But, for those who still feel they have been cheated by this post’s lack of “real math,” I give you two definitions which will lead us into the second post on this topic, which I hope to post in the next few days.

Definition 1. Let H be a subgroup of G, denoted H < G, and let g \in G, then the set gH=\{gh : h \in H \} is called the left coset of H (in G).

Definition 2. Let H < G. If \forall g \in G we have that gH=Hg, then H is said to be a normal subgroup of G, or simply “H is normal in G.” Normal subgroups are denoted by an left triangle, H \triangleleft G.