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Problem 73 (Every Finite Cyclic Group Of Order $n$ Is Isomorphic To $\mathbb{Z}_n$)

Let $H_n$ be the set of simple shift permutations on a set of $n$ letters so $H_n=\{\phi_k\mid 0\leq k\leq n\}$.

  1. Start by drawing the Cayley graphs of $H_7$ and $\mathbb{Z}_7$, just to make sure that the graphs are similar.
  2. Show that $f:H_n\to \mathbb{Z_n}$ defined by $f(\phi_k)=k$ is a group isomorphism. Remember, this means you must show that $f$ is a bijection (what's the inverse) and that $f(\phi_j\circ \phi_k) = (f(\phi_k)+f(\phi_j))\pmod n$.
  3. Show that if $G$ is a cyclic group of order $n$ with generator $a$, then the function $f:\mathbb{Z}_n\to G$ defined by $f(k)=a^k$ is a group isomorphism.

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