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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

A Talk Without Words:Visualizing Group Theory

Nathan Carter

Bentley College

January 7, 2008

cayley diagrams
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
slide10

Cayley diagrams of S4,

using two different sets of generators

slide11

Cayley diagrams of A5,

using two different sets of generators

family 5 abelian groups
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

example advanced topic sylow theory
Example advanced topic: Sylow Theory

Subgroups of S3:

S3 acting on them by conjugation:

for more information
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