Find the number of elements of order $1$, $2$, $5$ and $10$ in $D_{10}$.
I want to decompose the group into $\mathbb{Z}_2 \times \mathbb{Z}_{10}$ and then use my knowledge of cyclic groups, but I know this is incorrect because $\mathbb{Z}_2 \times \mathbb{Z}_{10}$ is Abelian and $D_{10}$ is not. I have seen some references to Sylow's theorem on the website, but we haven't learned this yet. How to proceed?
The answer to this was to recognize that all elements were rotations or order 2 flips. So count rotations as you would in $\mathbb{Z}_{2n}$, then add $n$ more order 2 elements (the reflections).