I think I'm a bit confused about the order of elements in cyclic groups.
If we suppose $G$ is a group of order 35, and let $x∈G$ such that x≠e, from Lagrange's Theorem $x$ will be of order 5, 7, or 35. If $x$ is of order 35, then $G$ is cyclic and thus has elements of order 5 and 7.
I don't understand the last part where it says that if $x$ is of order 35, $G$ has elements of order 5 and 7. I'd thought that if $G$ is cyclic then all elements of $G$ would have order 35. Obviously this is wrong, but I don't quite understand why?
Any help would be greatly appreciated!