Maybe Dedekind could have done what you are looking for. The sources for the early development apparently include Cauchy, Galois, Abel, Cayley, etc. However, according to Bell "Men of Mathematics" (enjoyable to read, but not a really scholarly work), page 282:
"This abstract point of view is that now current. It was not Cauchy's, but was introduced by Cayley in 1854. Nor were completely satisfactory sets of
postulates for groups stated till the first decade of the twentieth century."
Bell says the following on page 518:
"Dedekind was one of the first to appreciate the fundamental importance of the concept of a group in algebra and arithmetic. In this early work Dedekind
already exhibited two of the leading characteristics of his later thought,
abstractness and generality. Instead of regarding a finite group from the
standpoint offered by its representation in terms of substitutions [....],
Dedekind defined groups by means of their postulates [....] and sought to
derive their properties from this distillation of their essence. This is in the modern manner: abstractness and therefore generality."
I'm not sure if Dedekind is the "first decade of the twentieth century" mathematician Bell was talking about. I think a more interesting secondary question here is when did group theory explicitly include infinite and continuous groups. I'm guessing that in the early days, finite groups were the focus. The combination of the purely algebraic finite groups with analysis to study infinite groups was much later, and I guess Sophus Lie gets most of the credit for that.
