Abstract Algebra - Permutations I'm asked to show that $(1,2,3) \in S_3$ generates a subgroup which is normal.
I know that I could show it explicitly but that would be tedious. I think it may have to do with the fact that $(1,2,3)$ generates all even permutations but I'm sure there's something I'm missing. Any help would be appreciated.

I think that the standard way to do that is simply by brute force.
You have just to check 3 conjugations of the subgroup by the 3 elements outside of it, process that is neither hard nor long.
Anyway, a not-so-tedious way to do that is to show that it is the kernel of the (surjective) group homomorphism
$$\begin{aligned}\phi: S_3&\longrightarrow C_2\\
\sigma&\longmapsto \mathrm{sgn}(\sigma)
\end{aligned}$$
Source: Take a look at this Wikipedia page.
