What is the subgroup generated by $\{(12),(23)\}$

Question

I tried to use the definition of generated set as follow

The subgroup generated by $X$, $X$ subset of group $G$, is the intersection of all subgroups which contain $X$.

My question: if $S_3$ is symmetric group, what is the subgroup generated by $\{(12),(23)\}$?

I tried and I found the $S_3$ is the only group satisfies the definition.

Answer 1

$S_3$ is generated by all of its two cycles. These are $(12) $, $(13)$, and $(23)$. But $(23)^{-1} (12) (23) = (13) $, so $(13)$ is in the subgroup generated by $\{(12), (23)\} $, and so the subgroup generated is all of $S_3$.