Theory of homotopy: existence of a subgroup of parameters in homotopy classes of Lie groups.

Leave $ G $ be a group of lies and $ alpha: [0,1] a G $ A smooth path, connecting the neutral element. $ n_G $ of the group with a group element $ g $, that is to say $ alpha (0) = n_G $ Y $ alpha (1) = g $.

Can we find a homotopic subgroup of a single parameter? $ gamma: { Bbb R} a G $ Connecting the neutral element of the group with the group element. $ g $, that is to say $ gamma (0) = n_G $ Y $ gamma (1) = g $Y $ gamma $ prohibited for $[0,1]$ is in the same kind of homotopy $ alpha $?

If I have a homotopy class to connect a group element to the neutral element, can I choose a representative path in this homotopy class to be a group of a single parameter?