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Problem 37 (Three Similar Cayley Graphs From Different Contexts)

This problem asks you to draw 3 Cayley graphs. Each Cayley graph should only have 4 vertices.

  1. Draw a Cayley graph for the set $H_4$ of simple shift permutations of $X=\{1,2,3,4\}$ using the generating set $S=\{\phi_1\}$.
  2. Then draw a Cayley graph for the span of the permutation $(1,4,3,2)$, which we wrote in disjoint cycle notation.
  3. Then draw a Cayley graph for $\{f_a\mid a\in U(5)\}$ using the generating set $S=\{f_2\}$. Remember that $f_a$ is a permutation of $U(5)$ defined by $f_a(x)=xa\pmod 5$.
  4. What do you notice about these Cayley graphs?

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