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Find the number of elements of order two in the symmetric group $S_4$ of all permutations of the four symbols {$1,2,3,4$}.

the order two elements are two cycles.number of $2$ cycles are $6$.but the given answer is $9$.where I am wrong?can anybody help me.

more generally is there any formula for the problem "Find the number of elements of order $r$ in the symmetric group $S_n$"

amWhy
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priti
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2 Answers2

15

Hint: you're forgetting to include and count those permutations that are the product of two disjoint two-cycles in $S_4$:

$$(1\, 2)(3 \,4),\; (1\, 3)(2 \, 4),\; (1\, 4)(2 \,3)\; \in S_4,$$ which comprise three additional elements in $S_4$, each of order $2$, as are the six 2-cycles of order $2$ you counted.

Recall that the order of a permutation which is the product of disjoint cycles is equal to the $\;\operatorname{lcm}\;$ (i.e., the least common multiple) of the orders of its cycles.

amWhy
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  • yes.thanks alot. but can you explain the general problem please? – priti Feb 23 '13 at 08:36
  • priti - give me a little time to do this...(the general problem), as there is not specific formula for the number of elements of order $r$ in $S_n$, as it depends on n and whether r divides n, of course, and if so, for $n/r = d:$ exploring the prime factors of $d$, and how these factors partition r. – amWhy Feb 23 '13 at 14:59
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As any permutation can be written as a product of disjoint cycles, and the product of disjoint cycles has order equal to the least common multiple of those cycles' lengths (=orders), the group $\,S_n\,$ has elements of order $\,r\,$ iff we can find disjoint cycles in it with their lengths' l.c.m. equal to $\,r\,$.

Thus, for example, in $\,S_4\,$ the order of the element with largest order is $\,4\,$ (as distinct cycle decomposition like $(a, b, c, d)$) , whereas in $\,S_5\,$ we have elements of order $\,6\,$ , say $\,(12)(345)\,$ , etc.

What is left is to count the number of cycles in $\,S_n\,$ ...

amWhy
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DonAntonio
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