Are all the symmetric groups cyclic groups?
I know that by definition a cyclic group is a group that is generated by a single element. But if I've got a symmetric group like for example $H=\langle(2\ 1\ 3\ 4\ 5\ 6)(7\ 8\ 10\ 9) \rangle$ is it cyclic? I mean is it generated by one element or two? *($H$ is a subgroup of $S_{10}$)
Your group $H$ is cyclic since it is generated by one element. When you ask if all "symmetric groups" are cyclic groups, however, by the usual definition the answer is no. $S_2$ is cyclic, but $S_n$ for $n\geq 3$ are nonabelian and hence not cyclic. These are all of the symmetric groups.
It seems you are talking about subgroups of symmetric groups though, and the answer is still no. Besides the group itself, you can take $\langle (12),(34)\rangle$ in $S_4$. This is isomorphic to the Klein $4$-group and hence is not cyclic. I suspect that most subgroups of the symmetric groups for $n\geq 5$ are not cyclic, but I won't make such a bold claim without proof.