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Take the group of all Translation in .
Let's see how the generator of this group is the derivative operator:
The way to think about this is:
  • the translation group operates on the argument of a function
  • the generator is an operator that operates on itself
So let's take the exponential map:
and we notice that this is exactly the Taylor series of around the identity element of the translation group, which is 0! Therefore, if behaves nicely enough, within some radius of convergence around the origin we have for finite :
This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!