Are $D_{20}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$ isomorphic?

Question

Are $D_{20}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$ isomorphic?

For $D_{20}$ I am using the notation $D_{2n}$ and $\mathbb{Z}_{2}, \mathbb{Z}_{10}$ are additive groups.

I think not, and my justification is this: In group $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$ we have 2 elements of order 2, namely, $(1,5) $ and $(0,5)$; while in $D_{20}$ there are 10 rotations and 10 flips, where each flip has order 2, therefore they are not isomorphic. Is that so?

Answer 1

Since $\Bbb Z_{10}\times \Bbb Z_2$ is abelian and $D_{20}\cong \Bbb Z_{10}\rtimes \Bbb Z_2\cong\langle g,h\mid g^{10}, h^2, hg=g^{-1}h\rangle$ is not, the groups are not isomorphic.

Answer 2

$D_{20}$ has exactly $10$ reflections but $\Bbb Z_2\times\Bbb Z_{10}$ has only $\it{three}$ elements of order $2$, namely $(0+2\Bbb Z,5+10\Bbb Z),(1+2\Bbb Z,5+10\Bbb Z),(1+2\Bbb Z,0+10\Bbb Z)$, and thus $\require{cancel} D_{20}\;\cancel{\approx}\;\Bbb Z_2\times\Bbb Z_{10}$.