Thread: Isometries of the real line

1. Isometries of the real line

Let $\displaystyle N$ denote the group of isometries of a line $\displaystyle \mathbb{R}^{1}$. Classify discrete subgroups of $\displaystyle N$, identifying those that differ in the choice of origin and unit length on the line.

So I know the isometries of the line are translations and reflections, but I don't quite understand what the question is asking for. Obviously the composition of two distinct reflections in the line is a translation in the line, so I'm drawing a blank as to what the discrete subgroups are and what the "identifying..." part of the question is asking for. Any help would be appreciated, thanks.

2. Originally Posted by Pinkk
Let $\displaystyle N$ denote the group of isometries of a line $\displaystyle \mathbb{R}^{1}$. Classify discrete subgroups of $\displaystyle N$, identifying those that differ in the choice of origin and unit length on the line.

So I know the isometries of the line are translations and reflections, but I don't quite understand what the question is asking for. Obviously the composition of two distinct reflections in the line is a translation in the line, so I'm drawing a blank as to what the discrete subgroups are and what the "identifying..." part of the question is asking for. Any help would be appreciated, thanks.
Maybe I am misinterpreting you but is it possible that $\displaystyle N=\{\pm 1\}$? Assuming you're equipping $\displaystyle \mathbb{R}$ with the usual (unique up to a scaling factor) inner product then the only isometries are $\displaystyle L(x)=\pm x$...

3. Well, I know you can divide $\displaystyle N$ into isometries which have orthogonal part 1 and isometries which have orthogonal part -1. So translations are the form $\displaystyle f(x) = x + a$ and then you have reflections about a point. I'm just having trouble formalizing this or determining what the discrete subgroups are (compositions of two reflections are translations, etc.)

4. Originally Posted by Pinkk
Well, I know you can divide $\displaystyle N$ into isometries which have orthogonal part 1 and isometries which have orthogonal part -1. So translations are the form $\displaystyle f(x) = x + a$ and then you have reflections about a point. I'm just having trouble formalizing this or determining what the discrete subgroups are (compositions of two reflections are translations, etc.)
Hmm, I'm lost. Aren't you calling linear transformations isometries if they preserve norm? So every linear transformation on $\displaystyle \mathbb{R}$ has the form $\displaystyle L(x)=ax$ and so $\displaystyle 1=|1|=\left|L(1)\right|=|a|$ and so $\displaystyle a=\pm 1$. What more could you mean?

5. Well, at least from what I'm getting from Artin's book, he's talking about if we consider the isometries when the origin could be any point. An isometry of the line will carry the line onto itself and preserve lengths. A translation by some unit length vector carries the line onto itself, so it is an isometry of the line. The unit length vector can be any length and that would generate another subgroup of translations. Those subgroups I think I have. But then you have that you can reflect the line through any point on the line and it'll be an isometry since again, the line is carried onto itself. And again, we can use any point as a reflection point. But say you take a group with two reflection lines, then it's composition is a translation, and this is where I'm getting lost on what the discrete subgroups containing reflections look like.

6. hmm...i think that there is a bijection from the additive group of R to the translations:

a-->Ta, where Ta(x) = x+a.

the reflections...well we can describe a reflection about the point b as: Rb(x) = 2b - x.

note that Ra o Rb(x) = Ra(Rb(x)) = Ra(2b - x) = 2a - (2b - x) = x + 2a - 2b = T(2a-2b)(x).

Ta o Rb(x) = Ta(Rb(x)) = Ta(2b - x) = 2b+a - x = R(b+(a/2))(x)

Ra o Tb(x) = Ra(Tb(x)) = Ra(x + b) = 2a-b - x = R(a-(b/2))(x).

T0 is the identity for this group. every reflection Rb is its own inverse (so {T0,Rb} is a subgroup for every real number b). T(-a) is the inverse of Ta. the translations form a subgroup, which also contains subgroups isomorphic to (Q,+), (Z,+) and (nZ,+), as well as every other additive group of every subfield of R (there are a LOT of these).

what more would you like to know?

7. Thanks for the help.

What do the subgroups that are generated by a translation and reflection look like/are they discrete subgroups? And same question for the subgroup generated by two distinct reflections.