This is a closed simply connected solvable normal subgroup K of G such that G/K is reductive.
ii) X(T) = Hom (G, C) is a rational vector space, so T is simply connected [17, p.
If T and hence T/C is simply connected, then C is a direct factor of T [21, p.
0] is simply connected, and Q is a pro-torus with character group isomorphic to Hom (H, C).
1]) is a connected algebraic subgroup of the simply connected pro-torus T.
A connected pro-affine algebraic group G is called simply connected if every covering of G is an isomorphism.
rho]] of G by H exists and E/G is simply connected.
Since G' is a connected normal algebraic subgroup of H', and H' is simply connected, then G' and H/G' are simply connected.
rho]] is a [tau]-extension of G by H such that E'/G is simply connected.
1] of G', the proof is further reduced to the case where G is simply connected and reductive.
Then we know from Theorem 17 that E/A is simply connected.
i,[gamma]'] is evidently a [tau]'-extension of G' by H and (X x E)/[iota](G') is simply connected being canonically isomorphic to E/A.