  Elements of the Riordan group $\cal R$ over a field $\mathbb F$ of
characteristic zero are infinite lower triangular matrices which are defined in
terms of pairs of formal power series. We wish to bring to the forefront, as a
tool in the theory of Riordan groups, the use of multiplicative roots
$a(x)^\frac{1}{n}$ of elements $a(x)$ in the ring of formal power series over
$\mathbb F$ . Using roots, we give a Normal Form for non-constant formal power
series, we prove a surprising simple Composition-Cancellation Theorem and apply
this to show that, for a major class of Riordan elements (i.e., for
non-constant $g(x)$ and appropriate $F(x)$), only one of the two basic
conditions for checking that $\big(g(x), \, F(x)\big)$ has order $n$ in the
group $\cal R$ actually needs to be checked. Using all this, our main result is
to generalize C. Marshall [Congressus Numerantium, 229 (2017), 343-351] and
prove: Given non-constant $g(x)$ satisfying necessary conditions, there exists
a unique $F(x)$, given by an explicit formula, such that $\big(g(x), \,
F(x)\big)$ is an involution in $\cal R$. Finally, as examples, we apply this
theorem to ``aerated" series $h(x) = g(x^q),\ q\ \text{odd}$, to find the
unique $K(x)$ such that $\big(h(x), K(x)\big)$ is an involution.
