arXiv:1201.3741v2 [math.DG] 21 Jan 2012 G-set Theory and Applications in Lie Theory Reza Aghayan * Mehdi Nadjafikhah † January 24, 2012 Abstract This paper is devoted to the development and applications of some (new) basic concepts in Lie theory, both from “computational” and “ob- servability” viewpoint. We specify set of all “G-equivariant” maps from a given Lie group G to the underlying manifold M, namely G-set, and also we introduce “conjugacy” in Lie group theory. The next goal of this paper is detailed analysis of the G-sets in connection with underlying transformation groups and providing a rigorous theoretical justification of “G-sets”, when a group of transformations G acts on manifold M. Mathematica subject classification 2010: 22E15, 54H15, 58D15, 58D19. Keywords: Transformation groups, G-equivariant maps, G-invariance objects, Moving frames, Invariant-functions and Unique structure. * Kingston University, London, UK. E-mail: reza aghayan@kingston.ac.uk. † School of Mathematics, Iran University of Science and Technology, Narmak-16, Tehran, I.R. Iran. E-mail: m nadjafikhah@iust.ac.ir 1