1

If $G$ acts on $X$, then $X$ is the disjoint union of the distinct orbits. That is, one can write $$ X = \bigcup_{x\in R} orb_G(x) $$ where $R$ is a set of representatives, with one element from each orbit.

It might just be me, but this feels a bit awkward. Is there another way to say this "one element from each orbit"? Is there another way to properly define $R$?

John Doe
  • 3,233
  • 5
  • 43
  • 88

1 Answers1

1

Let $Y$ be the quotient of $X$ by the action of $G$, defined in the following way. Define an equivalence $\sim_G$ relation on $X$ by $$x_1\sim_G x_2 \iff \exists g\in G : x_2=g.x_1$$ and set $Y:=X/\sim_G$. (When the action is clear from context, the notation $X/G$ is also used).

For each $y\in Y$, set $O_y:= \mathrm{orb}_G(x)$ for any $x$ with $[x]=y$, where $[x]$ denotes the quotient projection (the class of $x$ in $Y$). This definition is well-posed by definition of $\sim_G$, i.e. if $x'$ is such that $[x']=y$, then $\mathrm{orb}_G(x')=\mathrm{orb}_G(x)$.

Now we have $$X=\bigcup_Y O_y\quad.$$

In words, $Y$ replaces $R$ in that we do not need to choose one representative $x$ for each class $y=[x]$, and can rather directly choose as indexing set the set of classes. The awkwardness in using $R$ comes precisely from the arbitrariness of the choice of a representative for each class, but we in fact we do not need to make such a choice in the first place.

AlephBeth
  • 1,752
  • Of course we can write $X$ as a (disjoint) union of the (partitioning) equivalence classes. But the OP is asking how to less awkwardly do so in representative language, so this is not an answer to that question. – Bill Dubuque Aug 25 '22 at 19:23
  • 3
    The purpose of an answer can be that to suggest an alternative way of thinking to the question. Of course one can write $X$ as a union indexed over a set of representatives, yet no terminological nuance will make the awkwardness of this description go away. Presenting a different, arguably more precise and canonical, way to describe $X$ (as union indexed by its quotient space) might be a polite way to suggest that using representatives may not be a good choice in the first place. – AlephBeth Aug 26 '22 at 01:04