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We consider formal power series $f(z) = \omega z + a_2z^2 + \ldots \ (\omega\neq 0)$, with coefficients in a field of characteristic $0$. These form agroup under the operation of composition (= substitution). We prove (Theorem 1)that every element $f(z)$ of finite order is conjugate to its linear term$\ell_\omega(z) = \omega z$, and we characterize those elements which conjugate$f(z)$ to $\omega z$. Then we investigate the construction of elements of order$n$ and prove (Theorem 2) that, given a primitive $n$'th root of unity $\omega$and an arbitrary sequence $\{a_k\}_{k\neq nj+1}$ there is a unique sequence$\{a_{nj + 1}\}_{j=1}^\infty$ such that the series $f(z) = \omega z + a_2z^2 +a_3z^3 + \ldots$ has order $n$. Sections 1 - 5 give an exposition of thisclassical subject, written for the 2005 - 2006 Morgan State UniversityCombinatorics Seminar. We do not claim priority for these results in thisclassical field, though perhaps the proof of Theorem 2 is new. We have now(2018) added Section 6 which gives references to valuable articles in theliterature and historical comments which, however incomplete, we hope will giveproper credit to those who have preceded this note and be helpful and ofinterest to the reader.

Elements of Finite Order in the Group of Formal Power Series Under Composition