Given order for every element in a symmetric group

[Mr Davis 97]

Compute the order of each of the elements in the symmetric group $S_4$.

Is the best way to do this just to write out each element's cycle decomposition, or is there a more efficient way?

 

[Orodruin]

Since the order of the elements are the same within each conjugacy class, I would just take one representative of each conjugacy class. But yes, I would do it by looking at the cycles of that representative.

 

[fresh_42]

Compute the order of each of the elements in the symmetric group $S_4$.

Is the best way to do this just to write out each element's cycle decomposition, or is there a more efficient way?

You can decompose the group:
$$S_4 \cong A_4 \rtimes \mathbb{Z}_2 \cong (V_4 \rtimes \mathbb{Z_3}) \rtimes \mathbb{Z}_2 \cong (\mathbb{Z}_2^2 \rtimes \mathbb{Z_3}) \rtimes \mathbb{Z}_2$$