Examples
Such one-parameter groups are of basic importance in the theory of Lie groups, for which every element of the associated Lie algebra defines such a homomorphism, the exponential map. In the case of matrix groups it is given by the matrix exponential.
Another important case is seen in functional analysis, with G being the group of unitary operators on a Hilbert space. See Stone's theorem on one-parameter unitary groups.
In his 1957 monograph Lie Groups, P. M. Cohn gives the following theorem on page 58:
- Any connected 1-dimensional Lie group is analytically isomorphic either to the additive group of real numbers, or to, the additive group of real numbers mod 1. In particular, every 1-dimensional Lie group is locally isomorphic to R.
Read more about this topic: One-parameter Group
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