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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Are primitive words universal for infinite symmetric groups?


Author: D. M. Silberger
Journal: Trans. Amer. Math. Soc. 276 (1983), 841-852
MSC: Primary 20B30; Secondary 03D40, 20B35, 20F10
MathSciNet review: 688980
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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\vert k \right\vert \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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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0688980-5
PII: S 0002-9947(1983)0688980-5
Article copyright: © Copyright 1983 American Mathematical Society