G acts on {$a,b,c,d,e$}. let $g,h\in G$ such that $g =$ ($a$ $b$ $c$ $d$ $e$), $h =$ ($a$ $b$)($c$ $d$ $e$).
Prove that G is not solvable. I know that $S_5$ is not solvable and I want to prove that G isomorphic to it. I also know that $[G:G ∩ A_5] = 2$ but I don't know how to continue from here.
Any ideas?
The order of the action of $g$ is $5$ and the order of the action of $h$ is $6$. By Lagrange's theorem a subgroup of $S_5$ containing both of these actions must have order divisible by both $5$ and $6$, and therefore $30$. That leaves $30, 60$ and $120$ as possible orders. The only one with $60$ is $A_5$, and it can't be $A_5$ since the action of $h$ is an odd permutation. So the only thing left is to show that we cannot have $30$ elements.
Note that if we apply first $h$ then $g$, we get the permutation $(a\,c\,e\,d)$ which has order $4$, and such an action cannot exist in a group of $30$ elements. Therefore the group must have $120$ elements.