Group Operations/ Group Actions 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. 
 A: 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.
A: One popular example is the action of the modular group $SL(2,\mathbb{Z})/\pm I$,acting on the upper half plane $\mathbb{H}^2$ by Moebius transformations. This gives many insights about the group itself. For the above example see the notes of K. Conrad on the modular group, or on $SL(2,\mathbb{Z})$, which gives plenty of interesting results on $SL(2,\mathbb{Z})$, using group actions.
A: When you have a group $G$ acting on a set $X$ (a so-called $G$-set), you get a homomorphism from $G$ into $S_X$, the symmetric group on $X$.
Any group acts on itself by left (or right) multiplication.
So you get a homomorphism from $G$ to $S_G$.
It turns out that the action is faithful,  which implies that the homomorphism is an embedding.
This is the content of Cayley's theorem:   any group  $G$ embeds in the symmetric group on as many letters as there are elements of $G$.
A: As an example, consider the action of a group $G$ on itself by conjugation: it realizes the pointwise stabilizers as elements' centralizers, the orbits as conjugacy classes, the kernel as the center $Z(G)$, and the orbit equation as the Class Equation.
