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adjoint group Algebra arcwiso connoctod aro tho bo tho botwoon Cartosian Chooso closed invariant subgroup closed sets closod commutative compact consisting of tho continuous and opon continuous mapping correspondence cosots covering group curvo define denote differontial difforontiablo dofino dotorminod equations fiold fixod givon havo Hence Henco homoomorphism honcc honce Honco tho idontity imago indopondont infinitosimal transformations intorval isomorphism Lie algebra Lie group limit point linear locally arcwiso mappod matrices n-hood n-space neighborhood non-singular obtain olomonts open set opon sots paramotor group provo real numbers realization roadily roal rogard rosult satisfy set containing shrinkablo simply connoctod socond soparablo space spaco spaoo subset subspaco Suppose systom thero tho circle tho form tho functions tho group tho imago tho mapping tho points tho solution tho sot tho transformation thon thoro oxists thoso topological group topological space undor uniquo valuo voctor volocity field whoro