  We use gauge theoretic and algebraic methods to examine sufficient conditions
for smooth points on the moduli space of flat connections on a compact manifold
and on the character variety of a finitely generated and presented group. We
give a complete proof of the slice theorem for the action of the group of gauge
transformations on the space of flat connections. Consequently, the slice is
smooth if the second cohomology of the manifold with coefficients in the
semisimple part of the adjoint bundle vanishes. On the other hand, we find that
the smoothness of the slice for the character variety of a finitely generated
and presented group depends not only on the second group cohomology but also on
the relation module of the presentation. However, when there is a single
relator or if there is no relation among the relators in the presentation, our
condition reduces to the minimality of the second group cohomology. This is
also verified using Fox calculus. Finally, we compare the conditions of
smoothness in the two approaches.
