How many $\alpha \in S_n$ are there with $\alpha^2 = 1$?
I want to use just linear algebra, I have done it using combinatorics but in class that does not work, can I get a little help? What I know is that
(i j) = (j i) and (i j)(k I) = (k l)(i j).
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1Hint: Try it for $S_3,$ and $S_4$. Essentially, you are looking for the number of permutations of order two: See if there is a pattern? – amWhy Nov 30 '18 at 21:44
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@amWhy I see that they used combinatorics there, is that the only way? – Juju9708 Nov 30 '18 at 21:50
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This is not a linear algebra problem. Ultimately, it is a group theory problem about the symmetric group $S_n$, and combinatorics is closely related, a tool, as you will. The question is not a linear algebra question. – amWhy Nov 30 '18 at 21:52
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Ohhh my bad, I thought it as a linear algebra problem beacuase I saw it on a book about it, but I got it, thank you! – Juju9708 Nov 30 '18 at 21:54
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Also see How many $\alpha \in S)n$ are such that $\alpha^2$ = 1 – amWhy Nov 30 '18 at 21:54