# 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?