Determine if the subgroup $\{(1),(15)(23),(12)(35),(13)(25)\}$ of $S_5$ is cyclic.
What kind of methods I have in order to check wheter some group is cyclic? The identity here is $(1)$, but this group is not generated by $(15)(23)$ since it has order $2$. So if I just look at the orders of each element and find one that generates the group and has order $3$ that's the generator and the group will be cyclic? This method doesn't seem to generalize quite well if I would have larger groups.
For a group of order $n$ to be cyclic, it must have an element of order $n$ (it being a generator). This is necessary and sufficient for finite groups, but not for infinite groups; consider $\Bbb Z\times \Bbb Z$, which is not cyclic yet has an element of infinite order.