Affine space $A^n$ and definition of difference.

Question

I'm not sure if this question would be more appropriate in Physics.SE, if so let me know.

I need help in understanding this quote from "Arnold - Mathematical Methods in Classical Mechanics" (This is my translation from italian):

The affine $n$-dimensional space $A^n$ differs from $\mathbb R^n$ for the fact that there's no fixed origin in it. The group $\mathbb R ^n$ acts in $A^n$ like the group of parallel transport: $$\mathbf a \rightarrow a+\mathbf b \quad (a\in A^n, \mathbf b \in \mathbb R^n, a+\mathbf b\in \mathbb R^n)$$(in this way, the sum beetween two points in $A^n$ is undefined, but the difference is defined as a vector of $\mathbb R^n$.)

I have different questions:

1. Why is $\mathbb R ^n$ referred as a "group"? Is it referred to $\mathbb R ^n$ being a group under the operation of sum?
2. What does that mapping mean? Also, am I right in suspecting that the bold character in first $\mathbf a$ is a typo?
3. In other places, I've seen the sum beetween a point $p\in A^n$ and a vector of $\mathbf v \in \mathbb R ^n$ defined as the unique point $q\in A_n$ such that $f(p,q)=\mathbb v$, where $f$ is the function that gives to $A^n$ the affine structure. If this is the case (and I'm not sure about it) how does this define the difference beetween two points?

Answer 1

This question seems perfectly on topic here.

The vector space $\mathbb{R}^n$ is a group under addition - you should check the axioms yourself if you haven't seen this before.

I agree that there is a typo in the mapping. This is a map $f\colon A^n\times\mathbb{R}^n\to A^n$ given by $f(a,\mathbf{b})=a+\mathbf{b}$. If you're familiar with the terminology of group actions, this is an action of $\mathbb{R}^n$ on $A^n$ (and because $\mathbb{R}^n$ is abelian, we don't have to worry about whether it is a left or right action).

Now, given two points $a,b\in A^n$, there is a unique $\mathbf{v}$ such that $f(a,\mathbf{v})=b$, and we define $b-a=\mathbf{v}$ (note that while $a,b\in A^n$, their difference is in $\mathbb{R}^n$).

Some Extra Detail: The reason this works is that $\mathbb{R}^n$ acts freely and transitively on $A^n$, meaning that if $f(a,\mathbf{v})=a$ for all $a\in A^n$ then $\mathbf{v}=0$, and that for all $a,b\in A^n$ there exists $\mathbf{v}\in\mathbb{R}^n$ with $f(a,\mathbf{v})=b$. Any set with a group acting freely and transitively on it is called a torsor for the group, and behaves like a "group without an identity" (compare to "vector space without an origin"). You can always define subtraction (or division if you're using multiplicative notation) in a torsor in this way; the difference $y-x$ of two elements of the torsor is the unique element of the group that acts by taking $x$ to $y$.