In Dummit & Foote's Abstract Algebra text, page 28 the following problem appears:

9. Let $G$ be the group of rigid motions in $\mathbb R^3$ of a tetrahedron. Show that $|G|=12$.

Apparently, I misunderstand something. In page 23 the authors define the dihedral group $D_{2n}$ with the same wording, "rigid motions":

For each $n \in \mathbb{Z}^+$, $n \geq 3$ let $D_{2n}$ be the set of symmetries of a regular $n$-gon, where a symmetry is any rigid motion of the $n$-gon...

Here they allow for the symmetries to be reflections, thus getting $|D_{2n}|=2n$. However, following that approach I find that the $G$ in problem 9 has order $|G|=24$.

Am I doing something wrong? Is there a mistake in the formulation of the problem?


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    $\begingroup$ I agree with you that D&F are using inconsistent terminology here. The best face one could put on it is that they use "rigid motion" to mean a distance-preserving map that also preserves orientation. (That is, no reflections.) Then the regular $n$-gon has $n$ rotational symmetries around its center point, and another $n$ around axes that lie in its plane, if you consider it as lying in $\Bbb R^3$ rather than $\Bbb R^2$. For the latter rotations, if you consider only their effect on the $n$-gon's plane, are not rotations but reflections. $\endgroup$ – MJD Mar 25 '14 at 14:08

Given a tetrahedron $X \subset \mathbb{R}^3$, $\{T \in O_3(\mathbb{R}) : T(X)=X\}$ is a group of order 24 (it's isomorphic to $\textrm{PGL}_2(\mathbb{F}_3)$), and $\{T \in SO_3(\mathbb{R}) : T(X)=X\}$ is a group of order 12 (it's isomorphic to $A_4$). It really does depend if you're allowing reflections or not...

To my knowledge, a rigid motion is just a transformation which is distance preserving (which all elements of $O_n(\mathbb{R})$ certainly are), so I would say it's a group of order 24.

EDIT: as noted below, rigid motions are orientation preserving, not just isometries, so the group in question is a subgroup of $SO_3(\mathbb{R})$ and hence of order 12.

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    $\begingroup$ rigid motions are isometries preserving orientation (i.e elements of $SO_n(\mathbb{R})$ ), so the group for tetrahedron is of order 12 instead of 24. ( ref: Euclidean group ) $\endgroup$ – achille hui Mar 25 '14 at 11:40
  • $\begingroup$ Aha, I thought they were just isometries, not necessarily orientation preserving; thanks for clearing that up. So the distinction here is symmetries versus rigid motions? The terminology then doesn't seem very consistent in the text the OP is referencing! $\endgroup$ – ah11950 Mar 25 '14 at 11:46
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    $\begingroup$ One possibility is for the regular $n$-gon, the rigid motions in the book include rotations in $3$-dimension space. If one rotate a planar figure around an axis in the plane for $180^\circ$, the effect is the same as if you reflect the figure with respect to that axis. $\endgroup$ – achille hui Mar 25 '14 at 11:54

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