In the book Programming bitcoin by Jimmy Song, the author provides an exercise for learning about finite fields:
for k = 1, 3, 7, 13, 18, what is this set in F(sub)19?
{k * 0, k * 1, k * 2, k * 3, .... k * 18}
Do you notice anything about these sets?
and then comments:
The answer to Exercise 5 is why we choose to use fields
with a prime number of elements.
No matter what k you chose, as long as it's greater than 0,
multiplying the entire set by k will result in the same set as you started with
If I multiply with e.g. 3 then I get {0, 3, 6, etc}, a different set
What does the author mean by "same set as you started with"?
It isn't a different set: you get the same elements back, but in a different order. It looks like $1$ and $2$ were skipped, for instance, but $3\cdot 13=39\cong1\pmod{19}$ and $3\cdot 7=21\cong2\pmod{19}$.
Evidently, multiplication by a nonzero $k$ gives a bijection from $\Bbb F_p\to\Bbb F_p$. (For $k$ is invertible in $\Bbb F_p$.)