Visualizing Cosets In A Cayley Graph

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Problem (Visualizing Cosets In A Cayley Graph)

The graph below is the Cayley graph of a group of order 12. A multiplication table for this group would consist of 144 entries. On this problem, we will visually construct the cosets of several subgroups of $G$ to illustrate how the cosets create a partition of $G$. Your goal is to understand visually the difference between right cosets $Hg$ and left cosets $gH$. You will need 4 colors to complete this problem.

1. Let $H=\left<a\right> = \{e,a,a^2\}$. Compute all the right cosets $Hx$ of $H$ for $x\in G$. You should obtain 4 different right cosets. Use the graph below on your left and color the vertices so that two vertcies are colored the same if and only if they are in the same right coset. Then repeat this with the left cosets $gH$ of $H$, and color your vertices in the graph on the right.
   
2. Now let $H=\left<b\right> = \{e,b\}$ and repeat part 1. Because $|H|=2$, you should obtain 6 cosets.
   
3. Now let $H= \{e,b,i,k\}$ and repeat part 1. Because $|H|=4$, you should obtain 3 cosets.
   
4. Now draw the 6 identification graphs obtained by considering either right cosets or left cosets for each of the subgroups above. You should notice that the only identification graph that is a Cayley graph is the last one where $H= \{e,b,i,k\}$. This happens precisely because the left and right cosets are the same, namely $gH=Hg$ for every $g\in G$. This is not a coincidence.

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