I had my first meeting for this independent study yesterday (Friday, Jan. 25), in which we discussed several exercises from chapter 2 of Gallian. (Chapters 0 and 1 focused primarily on fundamental prerequisites, like basic number theory and techniques of proof, as well as on an intuitive exposition of groups.) The exercises that we discussed assumed only basic knowledge of the definition of a group, some essential properties of groups, and understanding of one specific type of group (dihedral groups). We fully solved two exercises, and Dr. Prudhom gave me a hint on a third that I’m now trying to solve on my own.
The two exercises that we ended up solving entirely were numbers 17 and 39, which read as follows:
While I did learn the solutions to the exercises, which improved my understanding of groups, I found the process of solving the exercises to be most interesting. For 17, I had expected there to be some “trick” to solving the problem that made it quick and easy once you saw it, but it turned out that it just required some careful thought. In other words, we had to be willing to experiment and try things out in order to eventually solve the exercise. The same was true for 39; we spent a while trying various approaches to the problem until we finally encountered the correct one by looking online for hints. If I remember correctly, the hint that we found suggested finding some x such that axb=bxa in any, possibly non-Abelian group, from which it would follow from the premise of the exercise (specifically, that axb=cxd implies ab=cd) that ab=ba. Once we got the hint, it was just a matter of making educated guesses at what a suitable x would look like.
Finally, we also discussed a third exercise, which I’m now trying to solve using the hint that Dr. Prudhom gave me. This exercise was as follows:
The nth dihedral group is the group of symmetries of a regular n-gon. Before the meeting, I had made an educated guess that the only reflections that commute with each other are ones with perpendicular axes of reflection. If this turns out to be the case, then it follows that the composition of the two reflections is the same as a rotation by 180°. Dr. Prudhom suggested proving this not using group theory, but rather with coordinate plane geometry. That is, he suggested that I try to show that if the result of reflecting a point over the line y=mx and then over the line y=nx is the same as reflecting first over the line y=nx and then the line y=mx, then n=-1/m (which means that the lines are perpendicular). This general statement implies the group-theoretic version because we can think of the vertices of a regular n-gon as points on the coordinate plane.
Again, the approach that he suggested to this problem was quite interesting, and highlights how problems can be “translated” into an equivalent form that’s easier to solve. In this case, as opposed to solving the original group-theoretic problem, we translated it into a problem using basic algebra and geometry, topic with which I’m much more comfortable.