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I'm just starting my study of group theory, and I'm trying to figure out how to find every element in some order of a given group. Say I have the group $S_4$ (symmetric group of degree $4$). How could I find, say, all of the elements of order $3$, or $4$, or $5$? Is there some general process to find all elements for an order $n$?

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If you really want a general process, the general process is go through the group elements one-by-one, testing each to see what its order is. Well, this will work if the group is finite, anyway. There are more efficient methods, but you will come across them as you learn more group theory. – Gerry Myerson Dec 5 '12 at 2:37
up vote 2 down vote accepted

The elements of order $k$ in $S_n$ have cycles of lengths with lcm $k$.

For example: Elements of order 6 in $S_4$ would have to have either a 6-cycle, but there are no 6 distinct elements or at least an even cycle and a cycle whose length is divisible by three, but $2+3 > 4$, so this isn't possible, either. Therefore, there are no elements of order 6 in $S_4$

The elements of order 3 in $S_4$ are the 3-cycles, so there are $8$ such elements.

For groups whose order is divisible by a prime $p$ but not by $p^2$, the restriction on numbers of $p$-Sylow groups can be used to calculate the number of elements of order $p$.

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