Math 120A HW 5
Due Wednesday, February 11.
- 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 $n$.
- 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.
- 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.
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Prove that $\phi$ is a homomorphism.
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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 $\mathbb{Z}_n$.)
- 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.