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Problem 67 (SKIP) (Practice With Cyclic Subgroups)

In each part below, you are given a group. Find an integer $n$ so that the given group $G$ is isomorphic to $\mathbb{Z}_n$. If you were to create an isomorphism $f:\mathbb{Z}_n\to G$, what value would you assign to $f(1)$? The first one already has an answer.

  1. Let $G=\langle R_{270} \rangle$ as a subgroup of $D_{4}$, the automorphisms of a square.
    • We know that $R_{270}$ has order 4, so this subgroup is isomorphic to $\mathbb{Z}_4$. To obtain an isomorphism, we'd just let $f(1)=R_{270}$, as $R_{270}$ is a generator for $\langle R_{270} \rangle = \{R_{270},R_{180},R_{90},R_{0}\}$.
  2. Let $G=\langle R_{30} \rangle$ as a subgroup of $D_{24}$, the automorphisms of a regular 24-gon.
  3. Let $G=\left<\begin{bmatrix}2&1\\1&0\end{bmatrix}\right>$ in the general linear group $\text{GL}(2,\mathbb{Z}_3)$. [Hint: Just as all the previous problems, start computing powers of this matrix until you obtain the identity.]
  4. Let $G=\{(),(1,2,3), (1,3,2)\}$ as a subgroup of the set of all permutations of $X=\{1,2,3\}$.
  5. Let $G=U(7)$. [Hint: You'll need to start by finding a generator.]
  6. Let $G=U(17)$.

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