Regarding fixed points in finite groups of isometries

[fronton]

There is a theorem for finite groups of isometries in a plane which says that there is a point in the plane fixed by every element in the group (theorem 6.4.7 in Algebra - M Artin). While the proof itself is fairly simple to understand, there is an unstated belief that this is the only point that is fixed. Can somebody point me to a proof that there is only one point fixed by the group over all points in the plane? I would be so grateful,

Thanks,
Kind regards,
--
SACHIN

 

[algebrat]

... there is an unstated belief that this is the only point that is fixed. Can somebody point me to a proof that there is only one point fixed by the group over all points in the plane?

An unstated belief, hmm..., how do you know they have this unstated belief?

The group $M$ they refer to includes reflections, which fixes entire lines.

 

[lavinia]

A translation is an isometry that fixes no points.

 

[Vargo]

Yes, but a translation would not be part of a finite group of isometries.

A reflection across a line generates a group of order 2 and it fixes an entire line, so the statement you are making is not true.

Now suppose you are talking about orientation preserving isometries. If two points are fixed by every element of the group, then the line between those points must also be fixed (isometries must take lines into lines, as they are the distance minimizing paths). Isometries also preserve angles, so if the line m is fixed, with P on m fixed. Let n be the line through P perpendicular to m. Isometries preserve angles, and we are just in 2D. So either line n is fixed, or it is flipped. But the latter is impossible for an orientation preserving map. Now the same logic holds for every line perpendicular to m. Since they sweep out the whole plane, every point in the plane must be fixed.

 

[lavinia]

Right I misread the post.

 

[fronton]

Thanks everybody for the clarification. You're right I did forget about reflections. But just merely out of mathematical curiosity is there a way to prove for isometries involving rotations only, not involving reflections, that only a single point is fixed for all rotations of a finite group. The reason that I got interested in this is because the fixed point theorem proof takes a general point and proves that the centroid of all isometries of a finite group fixes the centroid. I thought then what happens if one considers a different point? Will the centroid of all isometries of that point also be fixed and will it be the same point as the first one? Anyway if somebody has the answer to this please do share it,

Thanks,
Kind regards,
--
Sachin

 

[fronton]

I am sorry please ignore my previous post. I didn't read Vargo's reply which proves very nicely what I had set out to prove,