Find the order of each element.

Question

Find $\mathbb{Z}_{15}^x$. Find also the order of each element in $\mathbb{Z}_{15}^x.$

So first I tackled $\mathbb{Z}_{15}^x$ which gave me : $\{1,2,4,7,8,11,13,14\}$

Now to find the order of each element.

$[1]$ has order 1.

$[2]$ has order 4 since $2^1 mod15 \equiv 2, 2^2 mod15 \equiv 4, 2^3 mod15 \equiv 8, 2^4 mod15 \equiv 1$

$[4]$ has order 2 since $ 4^1 mod15 \equiv 4, 4^2 mod15 \equiv 1.$

$[7]$ has order 4.

$[8]$ has order 2.

$[11]$ has order 2.

$[13]$ has order 4.

$[14]$ has order 2.

Are the orders correct for the elements?

Thank you

Answer 1

Well, the order of $[1]$ is $1$, of course; it's the identity element.

You are right about $[2]$ and $[4]$. Besides, since $[2]\times[2]=[4]$ and since $\operatorname{ord}[2]=4$, automatically you have that $\operatorname{ord}[4]=2$.

Answer 2

Since the group has $\phi(15)=8$ elements, the orders must divide $8$. So $15$ is impossible. Since the group is not cyclic (as we know from MSE here), there is no element of order $8$, so the orders of elements must even divide $4$, i.e., can only be $1,2$ or $4$.