# Given order for every element in a symmetric group

## Main Question or Discussion Point

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?

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Orodruin
Staff Emeritus
Homework Helper
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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
Mentor
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:
$$S_4 \cong A_4 \rtimes \mathbb{Z}_2 \cong (V_4 \rtimes \mathbb{Z_3}) \rtimes \mathbb{Z}_2 \cong (\mathbb{Z}_2^2 \rtimes \mathbb{Z_3}) \rtimes \mathbb{Z}_2$$