Making Algebra Less “Abstract”

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:

  1. Cyclic group of order 3
  2. Cyclic group of order 6
  3. Cyclic group of order 7
  4. Dihedral group of order 6
  5. Dihedral group of order 8
  6. Symmetric group of order 6
  7. Symmetric group of order 24
  8. Alternating group of order 12
  9. The quaternions
  10. (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:

  1. They have relatively small orders, so are relatively easy to understand.
  2. They represent a variety of different “types” of groups (note: I understand that there are no non-solvable groups).
  3. They will make the ideas of “subgroup,” “normal subgroup,” “quotient group,” “isomorphism,” and “homomorphism” easier to teach.
  4. 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:

  1. 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).
  2. 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.

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2 Responses to “Making Algebra Less “Abstract””

  1. Anonymous Says:

    I’d add in Z_2 x Z_2 so students don’t think that all abelian groups are cyclic. Plus the automorphisms of this group are isomorphic to S_3 (a nice exercise for a good student).
    –Jill D.

    • bretbenesh Says:

      Jill,
      This is a fantastic idea—I am sorry I missed it in my planning. I will give them something akin to two coins for the physical model.
      I appreciate the help!
      Bret

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