Math 120A HW 5

Due Wednesday, February 11.
  1. Let $G$ be a cyclic group of finite order $n$. We will show that $G \cong \mathbb{Z}_n$. Let $a$ be a generator of $G$ and define a function $\phi : \mathbb{Z}_n \to G$ by $\phi(i) = a^i$. Recall that we showed $\phi$ to be a bijection (in other words, the elements of $\langle a \rangle$, and therefore of $G$, are enumerated without repetition as $a^0,a^1,\ldots, a^{n-1}$.) Prove that $\phi$ is a homomorphism. Be careful: the operation on $\mathbb{Z}_n$ is $+_n$, not $+$, so it's not quite as easy as the infinite order case that we did in class. Your proof will need to use the fact that $a$ has order $n$.
  2. Let $p$ and $q$ be distinct prime numbers. How many elements of $\mathbb{Z}_{pq}$ are generators, and why? Hint: it might be easier to first count how many elements are not generators.
  3. Let $n,k \in \mathbb{Z}^+$. Define a function $\phi:\mathbb{Z}_n \to \mathbb{Z}_n$ by $\phi(a) = ka$. Here $ka$ means the $k^\text{th}$ multiple of $a$ in the sense of $\mathbb{Z}_n$, which is equal to $\underbrace{a +_n \cdots +_n a}_{k \text{ times}}$. However, note that we can equivalently define $ka$ in $\mathbb{Z}_n$ by multiplying $k$ and $a$ in the usual sense and only reducing modulo $n$ at the end.
    1. Prove that $\phi$ is a homomorphism.
    2. Prove that if $k$ is relatively prime to $n$, then $\phi$ is an isomorphism. (An isomorphism from a group to itself is called an automorphism of the group, so $\phi$ is an automorphism of $\mathbb{Z}_n$.)
  4. Recall that $S_3$ is the group of all permutations of the set $\{1,2,3\}$, under the operation of composition. (You can write composition as $\circ$, or use multiplicative notation.) List all elements of $S_3$ and compute the order of each element. Use the notation $\begin{pmatrix} 1 & 2 & \cdots & n\\ a_1 & a_2 & \cdots & a_n \end{pmatrix}$ for the permutation that maps $1$ to $a_1$, maps $2$ to $a_2$, and so on.
There are only 4 problems this week.