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We consider elements of finite order in the Riordan group $\cal R$ over afield of characteristic $0$. Viewing $\cal R$ as a semi-direct product ofgroups of formal power series, we solve, for all $n \geq 2$, two foundationalquestions posed by L. Shapiro for the case $n = 2$ (`involutions'): Given aformal power series $F(x)$ of finite compositional order and an integer $n\geq2$, Theorem 1 states, exactly which $g(x)$ make $\big(g(x), F(x)\big)$ aRiordan element of order $n$. Theorem 2 classifies finite-order Riordan groupelements up to conjugation in $\cal R$. Viewing $\cal R$ as a group of infinitelower triangular matrices, we interpret Theorem 1 in terms of existence ofeigenvectors and Theorem 2 as a normal form for finite order Riordan arraysunder similarity. These lead to Theorem 3, a formula for all eigenvectors offinite order Riordan arrays; and we show how this can lead to interestingcombinatorial identities. We then relate our work to papers of Cheon and Kimwhich motivated this paper and we solve the Open question which they posed.Finally, this circle of ideas gives a new proof of C. Marshall's theorem, whichfinds the unique $F(x)$, given bi-invertible $g(x)$, such that $\big(g(x),F(x))$ is an involution.

Elements of Finite Order in the Riordan Group and Their Eigenvectors