Laboratory Experiences in Group Theory

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Laboratory Experiences in Group Theory

• Ellen J. Maycock
• American Mathematical Society

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In this talk, I will

• Give a general description of using laboratories
  in an undergraduate group theory course
• Give an example using the software Finite Group
  Behavior

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Why use laboratories in group theory?

• Students can generate many concrete examples
  easily, and consequently the course seems less
  abstract.
• There is an opportunity for virtually every
  mathematics major to have a taste of research and
  discovery in a structured setting.

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Why? (continued)

• Group theory is new to all undergraduate
  students. They are building conceptual
  understanding at the same time they are learning
  the definitions and structures. An easy-to-use
  software supports this process.

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Format of Labs

• Before the Lab
• Pencil and paper work -- essential!
• In the Lab
• Students generate data.
• Students work in pairs.
• Dont expect deep conceptual understanding in
  this setting.

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Format of Labs (continued)

• After the Lab
• Students summarize and generalize from the
  data and formulate conjectures.
• Writing is crucial at this stage this is
  when the real learning occurs.
• Students should prove at least one proposition
  or theorem in the write-up of the lab.

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How labs fit into the organization of the class

• 6 8 labs in a semester-long course
• Students begin lab in a classroom setting, but
  finish up as homework.
• Labs cant take the place of classroom lectures
  and discussions.
• Conjectures can drive a classroom discussion or
  motivate a lecture.

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Class organization, (continued)

• Once students learn the basic concepts and
  structures, there is an opportunity for baby
  research projects using the software.

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Still available online from the MAA
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Laboratory Experiences in Group Theory

• Bundled with the software Exploring Small Groups
  (ESG)
• Published in 1996, based on materials I developed
  for my classes at DePauw University.
• Even though the lab ideas are still innovative,
  ESG no longer runs on many computers.
• The laboratories can still be used with another
  software

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Finite Group Behavior FGB

• by Bayard Webb and Ed Keppelmann
• Windows program for learning beginning group
  theory
• Available without charge at the web site
• http//unr.edu/homepage/keppelma/fgb.html

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Example using FGB

• Symmetries of a Square D4
• 0804 in group library

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Example, continued

• Rename elements
• Generate subgroups
• Construct quotient group
• Compare to other groups of order 8
• Construct endomorphism