Non-abelian group of order $2n$ I am in the begining of studying abstract algebra and I ran into the following problem from Herstein's book:
For any $n>2$ construct a non-abelian group of order $2n$. (Hint:  imitate the relations in $S_3$.)
My efforts: I was trying to realize the meaning of hint but my efforts were unsuccesful. We know that $S_3$ is non abelian group and $S_3=\{e,\ \phi,\  \psi,\  \phi\circ \psi,\  \psi\circ \phi, \ \psi^2\},$ where $$\phi:\{x_1,x_2,x_3\}\to \{x_2,x_1,x_3\} \quad \text{and} \quad \phi:\{x_1,x_2,x_3\}\to \{x_2,x_3,x_1\}$$
But I do not  know how to apply the structure of $S_3$ with order  $3!=6$ to the group $G$ with order $2n$ .
Can anyone show the detailed answer, please?
I did not find anything useful in this site.
 A: Here is what I take Herstein's hint to mean.
$S_3$ is generated by elements $\phi$ and $\psi$ satisfying the relations $\phi^2=1$, $\psi^3=1$ and $\psi\phi = \phi\psi^{-1}$. Herstein is quoted as writing: Hint: imitate the relations in $S_3$. This suggests that he intends you to examine a possible group generated by elements $\Phi$ and $\Psi$ satisfying $\Phi^2=1$, $\Psi^n=1$ and $\Psi\Phi = \Phi\Psi^{-1}$. Is there such a group? Does it have order $2n$? Is it noncommutative?
You already have clues to the answer, because in $S_3$, the group of permutations of $\{1, 2, 3\}$, you indicated that you know that the permutations $\phi = (1\;2)$ and and $\psi = (1\;2\;3)$ generate the group and (it can be checked that they) satisfy the relations. This might suggest to you that you try to check whether the permutations $\Phi, \Psi$ of $\{1,\ldots,n\}$ defined by $\Phi=(1\;2)$ and $\Psi=(1\;2\;\cdots\;n)$ generate a group satisfying the desired conditions. 
It turns out that they do not satisfy the relations. Here is where the comments to the question come in. It is better to choose $\Phi=(1\;n)(2\;n-1)\cdots(\lfloor\frac{n}{2}\rfloor\;\lceil\frac{n}{2}\rceil)$ and $\Psi=(1\;2\;\cdots\;n)$. This choice looks a little weird, but if you label the vertices of a regular $n$-gon with the numbers $1, 2, \cdots, n$, then $\Psi$ represents a $2\pi/n$ rotation and $\Phi$ represents a reflection through some axis of symmetry. 

Could you demonstrate entire and detailed sokution?
Since Fred Goodman already commented on your problem, let me direct you to his algebra book, which is free online. He writes about groups satisfying $\Phi^2=1$, $\Psi^n=1$ and $\Psi\Phi = \Phi\Psi^{-1}$ in Section 2.3.
