How do you find the order of a cyclic group?

Question

What is the order of the cyclic group generated by $(1 2 5)(3 4)$? What is the order of the cyclic group generated by $(1 2 5)(3 5)$?

I've looked through my notes and can't find notes on this and can remember how to solve this?

Any help please and thanks.

Answer 1

HINT: For $(1\,2\,5)(3\,4)$ note that the cycles are disjoint, so $\big((1\,2\,5)(3\,4)\big)^n=(1\,2\,5)^n(3\,4)^n$.

For the second, multiply out: $(1\,2\,5)(3\,5)=(1\,2\,3\,5)$ (or $(1\,2\,5\,3)$, depending on whether you compose permutations from left to right or from right to left).

Answer 2

For every $a,b \in S_n$ sush that $a$ and $b$ are disjoint cycles , $O(ab)=lcm(O(a),O(b))$ so $O((125)(34))=O((125))O((34))=6$ but $O((125)(35)))=O((1235))=4$