Describe the subgroups of S_5 generated by the 5-cycles

I'm new to Group theory and I'm just checking on my understanding. One example of 5-cycle is $$(1\ 2\ 3\ 4\ 5)$$. Hence, a subgroup generated by this 5-cycle consist of $$\{(1\ 2\ 3\ 4\ 5), (1\ 3\ 5\ 2\ 4), (1\ 4\ 2\ 5\ 3), (1\ 5\ 4\ 3\ 2), e \}$$, where $$e$$ is the identity element.

But what happens if it is generated by 2 5-cycles? E.g. $$<(1\ 2\ 3\ 4\ 5),(1\ 4\ 3\ 5\ 2)>$$. I start to get different cycles. One such element in this group is $$(1\ 5\ 3)$$.

Therefore, is there any generalizations I can obtain from the subgroups of $$S_5$$ generated by the 5-cycles?

How about if I were to extend the question to subgroups of $$S_6$$ generated by 6-cycles? Wouldn't it be more complicated to obtain some generalizations?

The five-cycles are even permutations. So they all lie in $$A_5$$ and the group $$G$$ generated by two "independent" five-cycles is a subgroup of $$A_5$$. By Sylow's third theorem, as $$G$$ has at least two Sylow $$5$$-subgroups, it has at least six Sylow $$5$$-subgroups. There are only six Sylow $$5$$-subgroups in $$A_5$$ so $$G$$ must be the group generated by all the $$5$$-cycles. Therefore $$G$$ is normal in $$A_5$$.....
• @Icycarus That's right. As you notice, $n$ being prime is also very convenient... – Lord Shark the Unknown Oct 9 '18 at 21:13