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Are all groups of order $2\cdot p$ abelian?

I would like to prove that Dihedral group $D(p)$ is the only non-abelian group of that order. $p$ is any prime $>2$.

Thanks for advice.

Tim Beno
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    Do you know the Sylow theorems and what a semidirect product is? That's the usual way to classify groups of order $2p$. (Hint: The $p$-subgroup is normal) – Milo Brandt Dec 18 '17 at 23:12
  • Checked it but I didn´t quite connect to the process yet. I will check it once more. Thank you – Tim Beno Dec 18 '17 at 23:15
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    It is pretty straightforward also without Sylow's theorems. If $p$ is an odd prime, by Cauchy's theorem there is an element $a$ of order $2$ and an element $b$ of order $p$, which generates a normal subgroup. If $a$ and $b$ commute the group is isomorphic to $\mathbb{Z}/(2p\mathbb{Z})$, otherwise it is isomorphic to $D_{p}$. – Jack D'Aurizio Dec 18 '17 at 23:28

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