One-parameter subgroup

A one-parameter subgroup of a Lie group $G$ is simply a Lie group homomorphism

φ : R \to G .

It is a special kind of curve through the identity.

There is a bijection between one-parameter subgroups and the tangent space at identity (the Lie algebra of $G$ ). This bijection is given by the exponential map.

Outline:

In one direction, given a one-parameter subgroup $φ$ , we take $φ^{'} (0)$ .
Conversely, given $V \in g$ , we consider the left-invariant vector field $V^{#}$ in $G$ . This relation is 1-1.
This vector field gives rise to a flow $φ (t, m)$ which is a one-parameter local group of transformations. Since left-invariant vector fields are complete (see proof here), we can take $g_{t} = ϕ (t, e)$ .
It can be shown that $φ (t, m) = m \cdot g_{t}$ (see here) and this assignment $t \mapsto g_{t}$ is the corresponding one-parameter subgroup.