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Problem (Identification Graphs Using Normal Subgroups Are Cayley Graphs)
Let $G$ be a group and suppose that $N$ is a normal subgroup of $G$. Let $a,b\in G$ and suppose that $ba\in Nc$, meaning we have an arrow from $Na$ to $Nc$ that is colored by $b$.
- Why does $Nc=N(ba)=(ba)N$? This should follow immediately from the properties of cosets and definition of normal. Be prepared to explain.
- Prove that if $x\in Na$, then $bx\in Nc$. (This shows that following arrow $b$ from anything in $Na$ will get you an element in $Nc$.)
- Extend your result to prove that if $y\in Nb$, then we also have $yx\in Nc$.
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