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[Group Theory] Constructing Cayley Graph from Given Relations

  1. Jun 24, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that there exists a group of order 21 having two generators [itex]s[/itex] and [itex]t[/itex] for which [itex]s^3 = I[/itex] and [itex]sts^{-1} = t^2[/itex]. Do this exercise by constructing the graph of the group.

    2. Relevant equations
    Based on the given relations, we have [itex]t^7 = I[/itex].

    3. The attempt at a solution
    Since ##s## and ##t## have periods of 3 and 7, respectively, I know that the graph can be based on either 3 heptagons or 7 triangles. The back of the book has a solution based on 7 triangles, but I would like to construct a graph based on heptagons for some much-needed practice. I see that I need three concentric heptagons to give the 21 elements of the group. However, I am having a hard time understanding how to connect the vertices of the heptagons to satisfy [itex]sts^{-1} = t^2[/itex]. I have had similar issues with the graphs of simpler groups which I solved by brute force. However, this group is complex enough that I do not want to do that. Is there some algorithmic way of seeing how the vertices must be connected by [itex]s[/itex]? If not, how do I go about figuring out the proper configuration?

    Last edited: Jun 24, 2013
  2. jcsd
  3. Jun 24, 2013 #2


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    You know that s has to connect a vertex of one heptagon to a vertex in a different heptagon. For sts-1 = t2 to hold, then going from one heptagon to the next, taking a step around in the t direction, then coming back to the first heptagon is the same as taking two steps around the original heptagon.
  4. Jun 25, 2013 #3
    I figured it out! For some reason, I didn't realize that it didn't matter which two vertices you connect first, since from there you derive the rest. Thanks!
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