# What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have problems with understanding why we view sets and operations abstractly.

Additionally, it can often be difficult to motivate why these topics are useful. One reasonable motivation that I've seen is RSA, but this generally compounds a tenuous grasp of number theory with a shaky understanding of group theory. This boils down to giving students who lack a mathematical background a reason to care about the topic.

What other strategies and motivations have people tried to (1) motivate how and why we think abstractly, and (2) why abstract algebra is useful in practice?

• Haven’t your students already worked directly from axioms in calculus or linear algebra? – Wrzlprmft Mar 15 '14 at 21:27
• @Wrzlprmft Calculus is usually done as a "compute this derivative/integral" course. Linear algebra isn't often a pre-requisite--but I'd be interested in the analogous question for linear algebra if they get there first. – adamblan Mar 15 '14 at 21:29
• I'm actually a little confused by this question. Do you want motivation for why students should study abstract algebra, or motivation for why students should be interested in the abstract reasoning component of an abstract algebra course? Your RSA example seems to be the former, while the rest of your question leans toward the latter. – Jim Belk Mar 15 '14 at 21:44
• @JimBelk This is a good point. I think I'm asking both. I will incorporate this into the original question. Thanks! – adamblan Mar 15 '14 at 21:48

## 4 Answers

When I introduce groups I first go over a very (e.g., 15 examples) long list of particular groups. I go over each example and verify the group axioms without naming it such. Then I present the problem of studying all of these examples together, motivated by simply saving labour. Instead of proving things again and again for each case, we'd like a single uniform formalism in which to study all of them. From there the road to the definition of abstract group is quite short.

A bit further into the course, when we get to polynomials, I explain that polynomials are an excellent algebraic way to encode combinatorics and therefore it is important in lots of places (since counting is fundamental). I present the students with the question "Can you design two (biased) six-sided dice such that the probability for each sum between $2$ and $12$ is equal". The negative answer can be obtained by considering the generating polynomials.

Of course, if such theorems as the Burnside Lemma in group theory is covered, then lots of real-life applications can be given. I also like to very briefly mention some of the applications of group actions in chemistry.

I also find that many students try to visualize what an abstract group is. This is impossible of course, and for students who never took any abstract course, and are so used to just computing or drawing graphs, this is a big psychological hurdle. I strongly advise, several times, not to even try to visualize. Rather, they need to understand that they need to develop a feel for the axioms and the proofs, of course, through practice. But don't visualize!

• This is wonderful. Thanks so much for your thoughts. :) – adamblan Mar 15 '14 at 23:12
• my pleasure @adamblan :) – Ittay Weiss Mar 15 '14 at 23:30
• Two simple examples to illustrate the saving of labour: I once attended a lecture Group Theory for Physicists, where overly restrictive axioms were given, requiring that the inverse should be both, a left and a right inverse and that the identity should be both, a left and a right identity. Thus some time was wasted showing both for some example groups. Also, some time was wasted on showing the associativity of groups whose elements were functions and whose operation was the composition of those functions. – Wrzlprmft Mar 16 '14 at 9:25
• Could you share the set of example groups you use? Just giving "made up" groups won't fly... – vonbrand Mar 26 '14 at 1:19
• Aren't the set of rotations around a point an example of a group? Or any of the other symmetry groups? These seem pretty visual to me, and really helped me understand the inverse and identity operations. – David Wees Mar 26 '14 at 2:24

In general, issues with abstraction and axioms can boil down to losing the ability to visualize what's going on. I'd suggest appealing to visual/tactile intuition as much as possible, especially in the beginning of the course. Present motivating examples of groups and other fundamental concepts that students can (quite literally) play with.

For instance, talk about the Rubik's Cube! Or symmetry groups of simple polygons, and fractals. Act out permutation groups in small (ahem) groups.

Also did some googling, and this looks interesting: PascGalois Project

I learned algebra from Hungerford's undergraduate text, with rings before groups. This is salient for your question in a few ways.

1. The examples are more concrete and more familiar. Students will have seen $\mathbb{Z, Q, R, C}$, and polynomials and matrices. That plus arithmetic modulo $n$ gives plenty of examples, all easier to explain than the symmetric or dihedral or alternating group on $n$ elements.
2. The variety of examples shows the advantages of a general theory that deals with all of them. This motivates "how and why we think abstractly".
3. The applications come sooner for number theory and cryptography. This shows "why abstract algebra is useful in practice."

I was introduced to abstract algebra through it's application to number theory, using Richman's "Number Theory: An Introduction to Algebra" (Brooks/Cole, 1971) as a textbook. Sadly, the book is long out of print (already was in 1984). Having "concrete" examples of rings, groups, ... was essential.