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Exercise (Practice With Order)

In each part below, you are given an element $a$ of a group $G$. Find the order of that element.

  1. Let $a= R_{270}$ which is an element of $D_{8}$, the automorphisms of a square.
  2. Let $a= R_{30}$ which is a 30 degree rotation and an element of $D_{48}$, the automorphisms of a regular 24-gon.
  3. Let $a=\left<\begin{bmatrix}2&1\\1&0\end{bmatrix}\right>$ 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 $a=(1,2,3,4,5)$ which is an element of the set of all permutations of $X=\{1,2,3,4,5\}$.
  5. Let $a=3$ as an element of $U(7)$.
  6. Let $a=2$ as an element of $U(17)$.
  7. Let $a=3$ as an element of $U(17)$.

Click to see a solution.

In each instance, we just have to compute $a^k$ for $k=1,2,3,\ldots$ until $a^k=e$. The smallest $k$ for which this occurs is the order of $a$. You should have obtained the answers 4, 12, 8, 5, 6, 8, 16. Make sure you can explain why for each.