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So, we know that in a Dihedral Group $D_n$, if $r$ represents counterclockwise rotation by $2\pi/n$ radians and $s$ is any axis of reflection, then the elements of reflection stand as follows:

{$s, rs,r^2s, ...... , r^{n-1}s$} .

In the image attached, there is an octagon . Suppose i consider the reflection axis $'s'$ as $ab$. Now, since $r$ stands for rotation by a degree of $360/8 = 45^o$, $rs$ must rotate $s$ by $45^o$ to obtain the $rs$ as $cd$. But, $rs$ is shown to be the axis passing through vertices $3$ and $7$ . Why is this paradox happening?

Instead of us calculating where vertices 3 and 4 go individually considering $r$ and $s$ , lets consider the cumulative effect of $rs$. Since, we have fixed the axis s, now, rs means, rotate this $s$ axis by $r$ radians in counter clockwise direction and then, since $rs$ is a reflection, reflect the original figure about the new axis described by $rs$. Thank you

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  • $\begingroup$ I take it $r$ is rotation clockwise through $\pi/4$. $r$ takes 3 to 4, then $s$ takes 4 to 3, so $rs$ fixes 3, so it's reflection in 3-7. $\endgroup$ – Gerry Myerson Mar 29 '14 at 8:14
  • $\begingroup$ So, if it's clockwise rotation, r should rotate s ( $ab$ here ) through $45^o$ and produce $rs$ as the axis joining the mid points of $4-5$ and $8-7$ . But, textbooks in such cases show $rs$ as $4-8$ $\endgroup$ – MathMan Mar 29 '14 at 8:18
  • $\begingroup$ $rs$ is $r$ followed by $s$. $r$ takes ab to mid45 - mid78, then $s$ takes that to cd, so it's reflection in 3-7, as you said earlier. But now you say the textbook says 4-8. That's consistent with $r$ being rotation counterclockwise. So: does the book say $rs$ is reflection in 3-7? or does it say it's reflection in 4-8? $\endgroup$ – Gerry Myerson Mar 29 '14 at 8:28
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    $\begingroup$ No. Why should $rs$ "yield a line of reflection"? $rs$ is a reflection, and the question is, in what axis is it a reflection. And the answer is, since it takes ab to cd, it can only be the reflection in 3-7. $\endgroup$ – Gerry Myerson Mar 29 '14 at 12:15
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    $\begingroup$ No. The cumulative effect of a composition of functions is the effect of doing the first function and then doing the second. And I have worked out for you what that cumulative effect is. $rs$ doesn't "give a new line of reflection"; $rs$ is a reflection, a reflection in 3-7. But we are repeating ourselves. $\endgroup$ – Gerry Myerson Mar 29 '14 at 22:25

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