TITLE: What, precisely, does Klein's Erlangen Program state?
QUESTION [42 upvotes]: People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, describing that this program is about relating algebra and geometry, about relating transformation groups of spaces (Lie groups) and different geometries, invariants, etc.
What I haven't been able to find is a precise statement in modern language (e.g. not that of his original paper) of what Klein's conjectures were. What precisely were his conjectures, or equivalently, what results constitute their resolution? Or was there never truly a precise statement?

REPLY [4 votes]: D. E. Rowe notes that one shouldn't read too much in program:

Klein wrote it to fulfill a formal requirement incumbant upon every new Ordinarius who entered the Erlangen faculty. Its name, in fact, had nothing to do with a new program for mathematical research, but rather derives from having been a “Programm zum Eintritt in die philosophische Fakultät und den Senat” at Erlangen.

(Likewise W. Killing and many others.)