Precise Definition of a Group Parameterization I've ran across the term "parameterization" used in group theory, especially for Lie groups, but I've never seen group parameterization precisely defined. 
From the context I usually find it in, it seems like a parameterization of a group $G$ is a mapping from a subset of a field to a representation of the group, where the subset is often defined by some finite number of constraints. For example, $SU(2)$ is parameterized by the subset of the field of quarternions obeying $qq^* = 1.$ It should be the case that $SO(3)$ can be parameterized by the same set, even though the mapping is only surjective in this case. Perhaps the latter is really an "inherited" parameterization since $SU(2)$ is the universal cover of $SO(3)$? 
Does anybody know of a precise definition of group parameterization? 
 A: Here is a tentative answer:


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*Mathematicians prefer not to think in terms of a parameterization (of Lie groups), but work instead with an atlas of charts, making parameterizations local. Among the charts, the most important one is at $e\in G$ and the standard chart at $e$ is given by the exponential map (although the choice of a  domain and codomain is ambiguous).  

*For Lie groups, as for general manifolds, there are other notions of "parameterizations" used in the literature. The only commonality between these notions is that a parameterization is required to be a smooth map whose image in $G$ has nonempty interior and contain $e$. Domains of such maps are sometimes allowed to be other manifolds, sometimes are required to be open subsets in ${\mathbb R}^n$. The maps themselves are sometimes required to be local diffeomorphisms, sometimes not.  Sometimes parameterizations are required to be injective/surjective, sometimes not. 

*The definition of a parameterization given in this Wikipedia article is OK as a non-technical explanation, but is lacking mathematical rigor. In contrast, the linked OSU page is essentially an explanation of an atlas of charts for non-experts.  
