The element of order 19

Question

I need to find the prime number $p\le11$ and number n $\in N$, such as multiplicative group of field of $p^n $ elements consist element of order $19$. May be I don't understand anything, but if the group of order $p^n$ must contain the element of order $19$, then by Lagrange theorem 19 must divide $p^n$, then $19$ divides $p$, then $p=19$, but according to the task $p\le11$. Can anyone help?

Answer 1

The multiplicative group of ${\mathbb F}_{p^n}$, i.e. ${\mathbb F}_{p^n}^*$, is cyclic with order $p^n - 1$. So you need to find $p$ and $n$ such that $19 \mid p^n - 1$, i.e. such that $p^n = 1 \pmod{19}$.

Answer 2

Hint: Note that $$ 5^9\equiv 1 \bmod 19. $$