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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?

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In the case of the pointwise stabilizer, reflection in the line itself also works. – Ben Blum-Smith Feb 21 '12 at 21:54
up vote 4 down vote accepted

Your two questions seem to be 1) Is your list complete? and 2) Is there a cleaner way to describe the group?

1) The list is missing glide reflections along the line.

2) There are probably many ways to express the group more cleanly, but what is coming to mind for me is to exhibit a list of generators rather than a list of the entire contents of the group. In this case, the group is generated by reflection in the line itself and the reflections in lines perpendicular to the line itself. All the other elements of the stabilizer can be expressed as finite compositions of these.

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Many thanks, do you know how could I express this as a generating set? Would it be something like $\langle F_{\alpha,u},F_{\beta_v} \rangle$? (Using notation from above) – Mathmo Feb 21 '12 at 22:11
Yes, that's exactly right. \langle, \rangle and commas. (But I think you meant F_{\beta,v} not F_{\beta_v}?) – Ben Blum-Smith Feb 21 '12 at 23:04
Yes I did, thanks. And great! – Mathmo Feb 21 '12 at 23:16

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