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Problem 69 (Cayley Tables And Isomorophisms On A Group Of Order 4)

In the problem Orders Of $\mathbb{Z}_n$ And $U(n)$ And Their Elements we computed the orders of $U(n)$ of $n$ from 2 to 10. The groups $U(5)$, $U(8)$, and $U(10)$ all have order 4 (they have four elements). For $U(5)$, we can construct a multiplication table (called a Cayley table) that represents the binary operation of the group. We create this table exactly as we created the times tables in grade school, where we put the product $ab$ in the row corresponding to $a$ and the column corresponding to $b$. The table for $U(5)$ is shown below. $$\begin{array}{c|cccc} \cdot \text{ mod } 5 & 1&2&3&4 \\ \hline\hline 1 & 1&2&3&4 \\ 2 & 2&4&1&3 \\ 3 & 3&1&4&2 \\ 4 & 4&3&2&1 \\ \end{array}$$

  1. Construct a multiplication table, i.e. Cayley table, for $U(8)$ and $U(10)$. Which of these groups is isomorphic to $U(5)$?
  2. Let $G = \{a,b,c,d\}$ and $H=\{r,s,t,u\}$ and define the operation $*$ on $G$ in the table on the left below, and the operation $\times$ on $H$ by the table on the right below. $$ \begin{array}{c|cccc} * & a&b&c&d \\ \hline\hline a & a&b&c&d \\ b & b&d&a&c \\ c & c&a&d&b \\ d & d&c&b&a \\ \end{array} \quad\quad\quad \begin{array}{c|cccc} \times & r&s&t&u \\ \hline\hline r & t&u&r&s \\ s & u&t&s&r \\ t & r&s&t&u \\ u & s&r&u&t \\ \end{array}.$$ For each group, which element is the identity element? Which group is isomorphic to $U(8)$?
  3. Let $K=\{1,-1,i,-i\}$, which is a subset of the complex numbers (recall that $i=\sqrt{-1}$ and $i^2=-1$). Construct a multiplication table for $K$, and show that $K$ is isomorphic to one of the groups above.

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