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I know there are two groups of order $34$. One is obvious $\mathbb{Z}/ (34\mathbb{Z})$, but what is the other? I know it must not be abelian. How do I go about describing it? Extending on Sylow theorems?

Stefan4024
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jacksonf
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    The dihedral group $D_{17}$ which has order $2\cdot 17=34$? (This is nonabelian) – Dave Nov 23 '17 at 19:31
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    Aut$(\mathbb{Z}{17})\cong\mathbb{Z}{16}$. Take an automorphism $\sigma$ of order $2$. Then the semi direct product of $\mathbb{Z}_{17}$ and $\langle\sigma\rangle\cong\mathbb{Z}_2$ has order $34$ and it is nonabelian (this argument can be applied to show that there exists a nonabelian group of order $pq$ when $p\mid q-1$.) – Levent Nov 23 '17 at 19:36

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By Cauchy's theorem, a group $G$ of order $2p$ where $p$ is an odd prime contains $C_p$ as a subgroup of index $2$. Any subgroup of index $2$ is normal, and $G/C_p \cong C_2$. We also know by Cauchy's theorem that $G$ has an element of order $2$, so the short exact sequence

$$C_p \to G \to C_2$$

splits. The conclusion is that $G$ is a semidirect product $C_p \rtimes C_2$. (This is a special case of the Schur-Zassenhaus theorem, and can also be done using the Sylow theorems, although that's overkill here.)

Now it remains to understand which automorphisms of $C_p$ have order $2$.We have $\text{Aut}(C_p) \cong C_{p-1}$, the cyclic group of order $p - 1$. This group has a unique nontrivial element of order $2$, namely $\frac{p - 1}{2}$ (times a generator); the corresponding automorphism is given by taking inverses in $C_p$. The conclusion is that there are exactly two groups of order $2p$, namely $C_p \times C_2 \cong C_{2p}$ (the semidirect product with the trivial action) and the dihedral group $D_p$ (the semidirect product with the unique nontrivial action, also known as $D_{2p}$ depending on your conventions).

Qiaochu Yuan
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