If $\sigma=(1\ 2\ 3\ 4)$, $\kappa=(1\ 2)$ for $S_4$ and I want to compute $(\sigma\kappa)^2$, does it become $\sigma^2\kappa^2 = \sigma^2$ (since $\kappa^2 = 1$), which is just $(1\ 3)(2\ 4)$?
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3"I want to calculate ()^2": well, just do it, and show your attempts (instead of trying to guess). – Anne Bauval Feb 09 '23 at 09:03
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Since $(\sigma\kappa)^2=\sigma\kappa\sigma\kappa$, asserting that $(\sigma\kappa)^2=\sigma^2\kappa^2(=\sigma\sigma\kappa\kappa)$ is the same thing as asserting that $\kappa\sigma=\sigma\kappa$. Well, this is not true. For instance, $\kappa\sigma$ maps $1$ into itself, whereas $\sigma\kappa$ maps $1$ into $3$.
Anyway, I think that the most natural approach for this problem is simply to compute $\sigma\kappa$ and then to square it.
José Carlos Santos
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Going with the approach of computing then squaring it, is this correct: = (1234)(12)(34) = (1)(2)(3)(4) ? If this is the right direction, how do I square (1)(2)(3)(4)? Thanks in advance – m1koto Feb 09 '23 at 09:58
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Where does the equality $\sigma\kappa=(1\ \ 2\ \ 3\ \ 4)(1\ \ 2)(3\ \ 4)$ come from? – José Carlos Santos Feb 09 '23 at 10:08
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Isn't =(1 2 3 4) and =(1 2)(3 4)? I might be misunderstanding reflections, how do I calculate reflections for symmetric group? – m1koto Feb 09 '23 at 21:44
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Oh Okay, so in this case it will be (1 2 3 4)(1 2)(3) (4)? Sorry I am a little confused by what reflections of symmetric groups mean algebraically – m1koto Feb 09 '23 at 22:25
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Yes, $\sigma\kappa=(1\ \ 2\ \ 3\ \ 4)(1\ \ 2)=(1\ \ 3\ \ 4)$. – José Carlos Santos Feb 09 '23 at 22:34
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In general, $(\sigma k)^2$ means $(\sigma k)(\sigma k)$, because you "multiply" the element $(\sigma k)$ two times with itself. (It's actually the group operation instead of multiplication).
In this case, $\sigma k\sigma k$ is not the same as $\sigma^2 k^2$, because the group $S_4$ is non-commutative.
Does this help?
student91
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