Hw4 Problem 2

### Problem 2

Give an example of a group which has exactly $231$ elements.

Solution
Consider the set $\mathbb{Z}_n$ under the binary operation $+$.
First, note that the set $\mathbb{Z}_n$ is closed under addition modulo $n$ by definition.
Next, let $a,b,c\in \mathbb{Z}_n$.
Then observe that $(a+b)+c=a+(b+c)$ by the associative property of addition.
Thus, $\langle \mathbb{Z}_n ,+ \rangle$ is associative.
Now, consider the elememt $0$ in $\mathbb{Z}_n$.
Let $a$ be any element in $\mathbb{Z}_n$.
Then $0+a=a+0=a$, by properties of addition.
Hence, $0$ is the identity for $\mathbb{Z}_n$.
Now, let $a\in \mathbb{Z}_n$.
Then note that $(n-a) \in \mathbb{Z}_n$, equaling $0$ when $a=0$.
Now consider $a+_n (n-a)=n-n=0$.
Similarly, observe that $(n-a)+_n a=n-n=0$.
Thus, for all $a\in \mathbb{Z}_n, (n-a)\in \mathbb{Z}$ is the inverse of $a$.
Therefore, by definition of group, $\langle \mathbb{Z}_n , + \rangle$ is a group.
$\hspace{300pt} \clubsuit$

Now, let $n=231$.
Then $\mathbb{Z}_{231} = \{0,1,2,...,230\}$ and has a total of $231$ elements.
Therefore, $\langle \mathbb{Z}_{231} , + \rangle$ is a group with exactly $231$ elements.
$\hspace{300pt} \clubsuit$.