Under mild conditions on p, closed subgroups of G(R) with full residual image G(F) are conjugates of G(A) for local subrings A, surjections arise from ring homomorphisms, and the identity is the universal deformation.
A structure theorem for subgroups of GL_n over complete local N oetherian rings with large residual image
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Deformation rings and images of Galois representations
Under mild conditions on p, closed subgroups of G(R) with full residual image G(F) are conjugates of G(A) for local subrings A, surjections arise from ring homomorphisms, and the identity is the universal deformation.