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Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length.

I know that:

  1. the conjugate of a reflection by a translation (or by any isometry, for that matter) is another reflection, by some explicitly calculation (haven't try).

  2. And 3 reflections leads to glide.

So I was thinking write a glide as 3 reflections and using the same process as proving 1 to prove the statement here. Is this promising? Can someone tell me? Or, should I try other methods?

Feel like this question is worth thinking. Thanks in advance~

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    $\begingroup$ What are your fundamental definitions for the various kinds of isometries? Knowing them might help decide what properties you can rely on and what you haven't shown yet. $\endgroup$
    – MvG
    Oct 1, 2013 at 19:09
  • $\begingroup$ Yes, your thinking is promising. $\endgroup$ Dec 11, 2015 at 22:58

2 Answers 2

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Since the Euclidean group is generated by reflections, it suffices to show that the conjugation of a glide reflection by a reflection is a glide reflection.

Let $\ell_{1}$ and $\ell_{2}$ be parallel lines, $\ell_{3}$ a line perpendicular to $\ell_{1}$, and $R_{i}$ the reflection across $\ell_{i}$, so that $$ G = R_{1} R_{2} R_{3} $$ is a glide reflection. (Conversely, every glide reflection can be represented in this way.)

If $R$ is an arbitrary reflection, then:

  • $R^{-1} = R$;

  • $RR_{i}R^{-1} = RR_{i}R$ is the reflection across the line $R\ell_{i}$. Denote this reflection by $R_{i}'$.

  • The lines $R\ell_{1}$ and $R\ell_{2}$ are parallel, and separated by the same distance as $\ell_{1}$ and $\ell_{2}$. The lines $R\ell_{1}$ and $R\ell_{3}$ are perpendicular.

Consequently, $$ RGR^{-1} = R(R_{1} R_{2} R_{3})R = (RR_{1}R)(RR_{2}R)(RR_{3}R) = R_{1}' R_{2}' R_{3}' $$ is a glide reflection having glide vector of the same length as $G$.

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    $\begingroup$ Since this had already gotten bumped.... $\endgroup$ Mar 15, 2016 at 12:44
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I think a conjugation of a glide reflection like rg$r^{-1}$, can be seen as rRt$r^{-1}$=rR$r^{-1}$rt$r^{-1}$, as the conjugation of reflection is still reflection, so does reflection, and conjugate will not change the direction, thus rR$r^{-1}$rt$r^{-1}$ is still a glide reflection.

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