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 $A_5$.....