TITLE: Group Operations/ Group Actions
QUESTION [5 upvotes]: I'm currently taking my first abstract algebra course and am learning about group actions, orbits, and stabilizers. I'm reading the Artin textbook and I am not very clear of what exactly a group action allows us to do, what it looks like, and why it's important. I know the two properties that must be satisfied to be a group action, but I just don't understand the usefulness of it yet. I have watched some videos of them and read a few other sections of some texts but am still not very clear. Does anyone have any simple clear examples to understand group actions, stabilizers, and orbits? Would be very much appreciated.

REPLY [5 votes]: Consider for example the action of $\Bbb Z\times\Bbb Z$ on the plane $\Bbb R^2$ given by $(m,n)(x,y)=(x+m,y+n)$.  The orbit of a point is a lattice in $\Bbb R\times\Bbb R$.  And the unit square $[0,1)\times[0,1)$ is a set of representatives, one point for each orbit.  The action identifies a side of that square with the opposite side.  Now if you take a square and identify opposite sides like that, you get a torus.  So the orbit space of this action is a torus.  Now having the torus as an orbit space allows us to identify certain structural properties of it and it gives us a nice continuous map from $\Bbb R^2$ to the torus obtained by mapping a point $p\in\Bbb R^2$ to its orbit.