I am teaching abstract algebra next semester, and I have decided to focus on
9 10 finite groups at the beginning of the semester. These groups are:
- Cyclic group of order 3
- Cyclic group of order 6
- Cyclic group of order 7
- Dihedral group of order 6
- Dihedral group of order 8
- Symmetric group of order 6
- Symmetric group of order 24
- Alternating group of order 12
- The quaternions
- (edit): The direct product of two cyclic groups of order 2 (thanks to Jill for recognizing my omission)
I have chosen these particular groups because:
- They have relatively small orders, so are relatively easy to understand.
- They represent a variety of different “types” of groups (note: I understand that there are no non-solvable groups).
- They will make the ideas of “subgroup,” “normal subgroup,” “quotient group,” “isomorphism,” and “homomorphism” easier to teach.
- Except for the quaternions, they all have physical representations for the students to study.
I have chosen to spend time concentrating on a handful of groups because:
- They will make abstract algebra less abstract. The physical representations will (hopefully) give the students a way of accessing the group structure. I have created physical representations for the students to use; see my website for details (note that I have borrowed—stolen, really—liberally from Patrick Bahls for the LaTeX section of this page. Also, my syllabus is only a draft in two senses: first, I might revise it more before classes start. Second, I intentionally let the students decide on many of the course policies, so the final draft will not be ready for another couple of weeks).
- Being very familiar with a handful of groups is the best way of producing counterexamples to conjectures; in particular, it seems like Alt(4) or the quaternions is the smallest counterexample for about 90% of false conjectures.
Of course, it is all just a hypothesis that having an intimate understanding of 9 finite groups will help students learn. I look forward to determining if the hypothesis is true.