Which of the following cannot be possible value for the number of elements of $G?$

Question

Suppose a finite group $G$ has an element $a$ which is not the identity such that $a^{20}$ is the identity. Which of the following cannot be possible value for the number of elements of $G?$

1) 12

2) 9

3) 15

4) 20

My work:Possible order for $a$ is $2, 4, 5, 10, 20 $ and we know order of element divides order of the group. From the above no one divide $9$. So option 2 is right answer.

Answer 1

The "formula" for checking this is just to check gcd:

Suppose $a^n=1$ and $G$ is a finite group. If $\gcd(n, |G|)=1$ then $a$ is the identity of $G$.

In your example, $\gcd(20, 9)=1$. As $a^{20}=1$ then $a$ is the identity. So you are correct - $|G|\neq9$.