# What is the connection between the generator of a Lie group, the element of the group, and the exponential?

(I'm not a mathematician, unfortunately, and I don't really master my own question. I have looked at all questions close to my question, but I could not understand anything because the level is much too high in mathematics.).

*Let's consider some generators g of a Lie group G. What are its elements ?

*If the group would not be a a Lie group, would the elements be different ones ?

*What is the connection between the generator of a Lie group, the element of the group, and the exponential function ?

*If the group is not a Lie group, is the expential disappearing ?

If $$G$$ is a Lie group with the Lie algebra $${\mathfrak g}$$ and $$\{X_1,...,X_n\}$$ is a basis of $${\mathfrak g}$$, then the elements $$X_1,...,X_n$$ are called generators of $$G$$. See for instance this answer.
1. Suppose now that $$\exp: {\mathfrak g}\to G$$ is the exponential map. Then for every element $$X\in {\mathfrak g}$$, $$\exp(X)$$ is an element of the group $$G$$. The same, of course, applies to generating sets: $$\exp(X_1),..., \exp(X_n)$$ are elements of $$G$$. Conversely, if $$g\in G$$ is sufficiently close to the identity element, then there exist $$X\in {\mathfrak g}$$ such that $$\exp(X)=g$$. In terms of the basis $$\{X_1,...,X_n\}$$, there are real numbers $$t_1,...,t_n$$ (if your Lie group is real) such that $$\exp(t_1X_1+...+t_nX_n)=g.$$ Lastly, if $$G$$ is compact and connected, then the above claim holds for all elements of $$G$$ regardless of how far they are from the identity element. This answers Question 3 about the relation.