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Problem 79 (Exercise - Solution Provided) (Practice With Identification Graphs On A Non Abelian Group)

Let $G$ be the automorphisms of the square, i.e. the dihedral group of order 8. Consider the subgroups $A=\left<R_{180}\right> = \{R_0,R_{180}\}$ and $B= \left<H\right> = \{R_0,H\}$. We'll use the set $S=\{R_{90}, V\}$ to draw the Cayley graph of $G$.

  1. Start by drawing the Cayley graph of $G$ with generators in $S$. You can check your answer with the Sage code at the end of this problem.
  2. For each $g\in G$, compute the right cosets $Ag$ of $A$. Draw the identification graph of $G$ using right cosets of $A$.
  3. For each $g\in G$, compute the left cosets $gA$ of $A$. Draw the identification graph of $G$ using left cosets of $A$.
  4. For each $g\in G$, compute the right cosets $Bg$ of $B$. Draw the identification graph of $G$ using right cosets of $B$.
  5. For each $g\in G$, compute the left cosets $gB$ of $B$. Draw the identification graph of $G$ using left cosets of $B$.
  6. Make a conjecture about what happens with identification graphs when $gA=Ag$. 
R90="(1,2,3,4)"
V="(1,4)(2,3)"

S=[R90,V]

G=PermutationGroup(S)
G.cayley_graph().show(color_by_label=True)
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