Subgroup of $S_5$ contains cycle of length $5$ and transposition

Question

Suppose a subgroup of $S_5$ contains a cycle of length $5$ and a transposition. Must it be all of $S_5$?

Say, it contains the cycle $(1 2 3 4 5)$ and a transposition $(ij)$. Then it contains a product of any number of copies of these two, and also inverses. How to continue?

Answer 1

Let $G$ is generated by $(12345)$ and $(ij)$. The $10\mid |G|$.

From reasons of symmetry it is sufficient to consider two cases:

1) $(12345)$ and $(12)$

2) $(12345)$ and $(13)$

In the first case $(12345)(12)=(1345)$ has order $4$ and $(12345)^2(12)=(14)(235)$ has order $6$. Hence $60\mid |G|$ and either $G=A_5$ or $G=S_5$. Since $G$ contains a transposition, $G\ne A_5$.

The second case is considered similarly.