[alphaBetaGamma]

Not really.

Very roughly: It's a group with a an infinite number of elements which is smooth, in the sense that there is a notion that elements can be close or far from each other, and locally around each element the group looks like R^n (for some n). Of course the group operation must play nice with this notion of "close". I.e. if elements A and B are close, after multiplying both of them by X, AX and BX can't be too far: the group operation does not "tear" the structure apart.

So in the example above of rotations on the plane, you can visualize the group as a circle where a point represents the rotation by that angle. Locally it looks like R, but it has a different global structure.