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It's been a while since I've dealt with these, but do character tables include all possible symmetry operations?

I am looking at the $D_{\infty h}$ point group table and it lists $C_\infty$, $\sigma_v$, $S_\infty$, $i$, and $C_2$. But doesn't it also have a $\sigma_h$? Why is this not included?

Am I missing something here?

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In the $D_{\infty \mathrm{h}}$ point group, the $\sigma_\mathrm{h}$ reflection is equivalent to the $2S_{\infty}(\phi)$ improper rotation with the rotation angle $\phi$ equal to zero (recall that the definition of a $C_n$ symmetry element is a rotation through $(360^\circ/n)$ degrees).

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  • $\begingroup$ Ah I didn't see that. Perhaps I'm being lazy and should be a different question, but why the "2" on "$2S_\infty$? $\endgroup$ – ralk912 Sep 4 at 2:35
  • $\begingroup$ I’m not sure whether it’s to do with the fact that you can rotate in either direction. Maybe ask a new question ;) I’m starting to doubt my answer, in fact, because the $C_n$ operation with $\phi = 0$ is also equivalent to $E$, and $E$ is one of the operations in the character table, so why not $\sigma_\mathrm h$. These infinite point groups are rather confusing... $\endgroup$ – orthocresol Sep 4 at 10:29
  • $\begingroup$ I assume you are right about you explanation and that the reason they separate out $E$, despite it technically being redundant, is that it makes explicit that $D_{\infty h}$ is a group, with the required identity operator. Why that is more important than making explicit that the group actually has a $\sigma_h$ operation is up to the judgement of whoever printed these tables. $\endgroup$ – Tyberius Sep 4 at 15:56
  • $\begingroup$ But also $i$, isn't $S_2$ equivalent to $i$? Or is $i$ particularly special too? $\endgroup$ – ralk912 Sep 5 at 18:01
  • $\begingroup$ @ralk912 yes, S2 is equivalent to i. I think you may have to dig deeper than my answer, and probably return to the formalism of group theory, to get to the root of the question - in particular, how it deals with infinite groups. I’m sorry I can’t help further right now. $\endgroup$ – orthocresol Sep 5 at 21:10

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