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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Are primitive words universal for infinite symmetric groups?
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by D. M. Silberger PDF
Trans. Amer. Math. Soc. 276 (1983), 841-852 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.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 276 (1983), 841-852
  • MSC: Primary 20B30; Secondary 03D40, 20B35, 20F10
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0688980-5
  • MathSciNet review: 688980