Why are isotropy groups, also known as stabilizers, named as such?
In physics, the word isotropy means having the same property in all directions. Can one draw an analogy from this to interpret the definition of stabilizers so that they indeed seem "isotropic"?
Think of a geometric context, like say rigid motions acting on the Euclidean plane. Then the stabilizer of a point is the group of rotations around that point. The point is fixed, but all the different directions coming out from that point are rotated into one another. So the isotropy group lets you compare what's happening in all the different directions coming out from that point.