My first time writing the quotient of a set of vectors by a matrix. Let the set of vectors $X=\pmatrix{x\\1}$ have $x\in\Bbb Q_2$ drawn from the 2-adic numbers.
Now let the matrices of the form $G=\pmatrix{2^\mathbb Z & \mathbb Z[\frac12]\\0&1}$ represent a group action $G$ expressed by the map:
$(t,u) \mapsto \pmatrix{2^u & t\\0&1}$
Question
How do I write the quotient of $X$ by the group action of $G$?
And how do I identify some element of:
- $G$
- $X/G$?
My Attempt (using $-\frac13$ as an example)
I guess the quotient is written as simply as $X/G$
I think an element of $G$ is simply written $\pmatrix{1&-\frac13\\0&1}$ *Note
An element of $X/G$ I'm not so sure about. Let's take for example the element $\pmatrix{\Bbb Z[\frac12]-\frac132^\Bbb Z\\1}$. I think that's the best notation for that, that I can suggest just now. Any improvements would be appreciated.
I guess something writing it as a single representative would be more elegant, like $G\pmatrix{-\frac13\\1}$
Or maybe $\pmatrix{2^\mathbb Z & \mathbb Z[\frac12]\\0&1}\times\pmatrix{-\frac13\\1}$
*Note: I'm aware $-\frac13$ isn't an element of the group. One must specify the power of 2 to which one is multiplying $x$ as well as the addor, in this case I've used $2^0$
The quotient of a set $X$ by a group action $G$ is commonly written $X/G$.
$X/G$ is the orbit space of the group action $G$ in the set $X$ and comprises the set of all orbits. If there's precisely one orbit, the group acts transitively on $X$.
In the example given above, the orbit through $\pmatrix{-\frac13\\1}$ can be written $G\cdot\pmatrix{-\frac13\\1}=\left\{g\cdot\pmatrix{-\frac13\\1}:g\in G\right\}$.