I have what appears to be a simple question but I do not know if my method is correct. I have to find what the subgroup of $S_5$ generated by $(123)$ and $(12345)$ is.
So I have worked out that the product of the above permutations is $(52134)$ so now do I compute the 5th power of the product I calculated? Then somehow find out what subgroup that is?
Since your subgroup will be of $\;A_5\;$ and this is a simple group, you are somehow restricted in your choices. For example, its index can't be less than $\;4\;$ as then the regular action would provide a contradiction.
But then you already have elements of order $\;3,\,5\;$ so at least you have a subgroup of order $\;15\;$...and this subgroup's index is already four, which is impossible, so it must be the subgroup is the whole $\;A_5\;$