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I just have a simple question regarding the definition of a Lie group. My book defines it as a group which is also a smooth manifold (e.g. its elements can be smoothly and continuously parameterized), and the group operation induces a smooth map of the manifold into itself.

I understand the first part of the definition, but I'm having trouble understanding the second half. What does it mean for a group operation to "induce" a map, and what exactly does it mean to have a smooth map of a manifold into itself? I mean I can picture a map which maps a manifold into itself, like a map from R->R (a simple function f(x) should do the trick right), but why impose that restriction? My book says something about making the group structure compatible with the structure of a manifold, but I didn't quite understand that either. Some help would be nice, thanks. =]