The natural semidirect product Rn⋊G(n) is algebraically determined

Note that, such a result is not true for Cn⋊GL(n,C) nor even for R3⋊SO(3,R). In fact, the proof of the theorem is done in a sequence of steps designed to verify the four hypotheses of the road map Theorem 2. In addition, one of the key intermediate results that requires a fair amount of labor is that φ−1(SO(n,R)) is an analytic subgroup of L, where SO(n,R) is the n×n rotation group with determinant one and SO(n,R)⊂SL(n,R)⊂G(n) for every n≥2.