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A Talk Without Words: Visualizing Group Theory

A Talk Without Words: Visualizing Group Theory. Nathan Carter Bentley College January 7, 2008. Cayley Diagrams. A graph whose nodes represent group elements and whose arrows represent the action of group generators Advantage: You can see the whole group and its structure simply.

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A Talk Without Words: Visualizing Group Theory

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  1. A Talk Without Words:Visualizing Group Theory Nathan Carter Bentley College January 7, 2008

  2. Cayley Diagrams • A graph whose nodes represent group elements and whose arrows represent the action of group generators • Advantage: You can see the whole group and its structure simply

  3. Family #1: Cyclic Groups

  4. Family #1: Cyclic Groups C3 Cn C5

  5. Family #2: Dihedral Groups

  6. Family #2: Dihedral Groups

  7. Families #3 and #4: Symmetric and Alternating

  8. Families #3 and #4: Symmetric and Alternating

  9. Cayley diagrams of S4, using two different sets of generators

  10. Cayley diagrams of A5, using two different sets of generators

  11. Family #5: Abelian Groups • Abelian groups are those in which all elements commute, ab=ba for any a and b in the group. Abelian Nonabelian

  12. Cayley diagrams for some example abelian groups

  13. Direct Products

  14. Direct Products

  15. Direct Products

  16. Direct Products

  17. Subgroup: A Little Group in a Big One

  18. Left Coset Right Coset Copies of a Subgroup: Cosets gH Hg

  19. Normal Subgroups

  20. Normal Subgroups

  21. The Quotient Process

  22. The Quotient Process

  23. Example advanced topic: Sylow Theory Subgroups of S3: S3 acting on them by conjugation:

  24. For More Information • Get Group Explorerhttp://www.platosheaven.com • Read about Group Explorer • Built-in and on-line documentation • Article in JOMA (link from GE home page) • Look for Visual Group Theoryin the second half of 2008 from the MAA

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