# Action of Isometries on a Line in the Plane

I'm trying to determine the stabilizer of a line in a plane when acted upon by the group of isometries of the plane. Please note that I'm using the notation found in the Wikipedia article on Euclidean plane isometries.

I've identified (and hopefully exhausted) the following isometries that will 'fix' a line $l$ set-wise. For the point-wise case, clearly only the identity transformation will work.

1. The identity transformation
2. Rotation about a point on the line by $\pi$ radians, i.e. $R_{c,\pi}$ for any point $c$ on the line $l$. I think this can also be expressed as $R_{c,\pi}=T_cR_{\pi}T_{-c}$
3. Translation in a direction parallel to the line, i.e. $T_v$ where $v$ is a vector parallel to $l$
4. Reflection about $l$ itself, i.e. $F_{\alpha,u}$ for any $\alpha$ lying on $l$, $u$ perpendicular to $l$
5. Reflection about any line perpendicular to $l$, i.e. $F_{\beta, v}$ for any $\beta$ in the plane, with $v$ parallel to $l$

I can't think of any more, but if I've missed any out, I'd be very grateful to learn of them. This leads me into my questions:

How can I 'formally' express the response to the given question:

The group G of isometries of the plane acts on the set of lines in the plane. Determine the stabilizer of a line in the plane.

Is there a better way of responding than just listing the isometries, as I have done here? i.e. Can we notate the set of the required isomteries more neatly?

Also,

How can I express reflections of the form $F_{\gamma,w}$ for some point $\gamma$ and some vector $w$ in the plane in terms of translations by some vector, rotations about some point and reflections about the $e_1$ axis? I book I'm working through by Artin states that any isometry $m$ can be written in the form $m=T_aR_{0,\theta}$ or $m=T_aR_{0,\theta}r$ where $r$ is a reflection in the $e_1$ axis.

Many thanks to anyone who can help. Best,

• My gut feeling is that this question is too elementary for this site, which is for questions of interest to research mathematicians. You might consider posting to math.stackexchange.com. If I were assigning this problem, I'd remark that isometries $f$ can be written in the form $f(x) = Mx + b$ for an orthogonal matrix $M$ and vector $b$, and remark that since the group acts transitively, we can choose any old line, say the $x$-axis, and see which $f$ fix this setwise. This puts restrictions on $M$ and $b$ (find them!). Any other stabilizer subgroup will conjugate to the one you just found. – Todd Trimble Feb 21 '12 at 19:20
• I disagree with "For the point-wise case, clearly only the identity transformation will work." What about reflection in your line (item 4 on your list)? – Andreas Blass Feb 21 '12 at 22:36

## 1 Answer

You seem to be missing glide reflections. For any translation along a line you can reflect across the line getting a glide reflection. They are in the wikipedia article you sited.