I'm not sure how to approach this question, and I have a guess:
What are all the possible orders of elements of dihedral group D17?
Since the order of any element in a finite group must divide the order of the group, and the order of D17 is 34, it follows any element of D17 is of order 1 (the identity), 2, 17, or 34.
Is this correct and how I find the possible orders of elements of any finite group?
Best Answer
You have the right idea! For sure, we know that if $g \in G$ for some group $G$, that the order of $g$ (denoted $|g|$) divides the order of $G$ (denoted $|G|$). So what you've written is that $1,2,17,34$ are all $\underline{\text{possible}}$ orders of elements of $G$.
However, this does not mean that all divisors of $G$ will have an element of that order; as anon pointed out, we know that $34$ is a divisor of the order of $D_{17}$, but that as $D_{17}$ is not cyclic there can't be an element of order 34.
To sum up:
$\left(g \in G\right)$ $\Rightarrow$ $\left(|g| \text{ divides } |G|\right)$
$\left(n \text{ divides } |G|\right)$ $\not\Rightarrow$ $\left( \text{There is some } g \in G \text{ with } |g| = n \right)$