Are primitive words universal for infinite symmetric groups?
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 by D. M. Silberger PDF
 Trans. Amer. Math. Soc. 276 (1983), 841852 Request permission
Abstract:
Let $W = W({x_1}, \ldots ,{x_j})$ be any word in the $j$ free generators ${x_1}, \ldots ,{x_j}$, and suppose that $W$ cannot be expressed in the form $W = {V^k}$ for $V$ a word and for $k$ an integer with $\left  k \right  \ne 1$. We ask whether the equation $f = W$ has a solution $({x_1}, \ldots ,{x_j}) = (a_{1}, \ldots , a_{j}) \in G^{j}$ whenever $G$ is an infinite symmetric group and $f$ is an element in $G$. We establish an affirmative answer in the case that $W(x,y) = {x^m}{y^n}$ for $m$ and $n$ nonzero integers.References

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Additional Information
 © Copyright 1983 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 276 (1983), 841852
 MSC: Primary 20B30; Secondary 03D40, 20B35, 20F10
 DOI: https://doi.org/10.1090/S00029947198306889805
 MathSciNet review: 688980