Roots in the substitution group and the Riordan group with unit diagonal

We investigate the existence and uniqueness of iterative roots of order n within the substitution group of formal power series J ( Z ) -with coefficients in a commutative ring with unity Z - employing a matrix-based framework grounded in the Riordan group. We analyse the relationship between the substitution group and the Lagrange subgroup - a group of Riordan matrices - and explore some classic questions concerning algebraic completeness and uniqueness of root extractions. This approach allows us to obtain various results about the roots in J ( Z ) for different choices of Z . Furthermore, the examination of the substitution group facilitates the analysis of roots within the Riordan matrix group.