Groups of order $34$ 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?
 A: 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). 
