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Problem (Subgroups Cosets And Identification Graphs Of The Automorphisms Of The Square)
Let $G$ be the automorphism group of the square, so the dihedral group of order 8.
- Make a list of all the subgroups of $G$. You should have 1+5+3+1=10, namely 1 of order 1, 5 of order 2, 3 of order 4, and 1 of order 8. Think about the game "generate/don't generate" and build these subgroups by spanning elements of $G$.
- For each subgroup $H$, make a list of the right cosets of $H$. Then construct an identification graph of $G$ using the right cosets of $H$. What relationship is there between $|G|$, $|H|$, and the number of vertices in the identification graph?
- A Cayley graph has exactly one arrow of each color leaving each vertex. Some of your graphs above satisfy this property, and some have more than one arrow of each color leaving each vertex. Go through your list and decide which cannot be Cayley graphs and and which can be.
- Pick one of the graphs that is a Cayley graph (preferably not $H=\{e\}$ nor $H=G$). For that subgroup $H$, compute the right cosets of $H$.
- Pick one of the graphs that is not a Cayley graph. For that subgroup $H$, compute the right cosets of $H$.
- Is there a connection between left and right cosets and when identification graphs are Cayley graphs? What do you notice. Check if you are correct by looking at another subgroup of $H$.
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