Permutions that commute

Question

So I have two permutations for which i worked out the product disjoint cycles.

f = (1 2 4)(3)(5) and h = (1)(2 4)(3)(5), i also worked out the order of both,

but how can you show that they commute if they do at all.

I know this is basic, Im just a beginner.

Answer 1

You don't have to calculate the complete composition but just look at the image of one element, see if it gives a difference and end, or take another point if equal. In your example start with $1$, then $f(1) = 2$ and $h(2) = 4$, on the other hand $h(1) = 1$ and $f(1) = 2$ so $hf(1) = 4$ and $fh(1) = 2$, So $fh \neq hf$.

Answer 2

Just compute $fh$ and $hf$ and see if they are equal.