1983
DOI: 10.2307/1999086
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Are Primitive Words Universal for Infinite Symmetric Groups?

Abstract: Abstract. Let W= W(xx,...,Xj)bc any word in thej free generators xx,...,Xj, and suppose that W cannot be expressed in the form W = Vk for V a word and for k an integer with \k\¥" 1. We ask whether the equation f=W has a solution (xx>... ,Xj) = (a,,... ,Oj) €E GJ whenever G is an infinite symmetric group and/is an element in G. We establish an affirmative answer in the case that W(x, y) = xmy" for m and n nonzero integers.

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Cited by 10 publications
(13 citation statements)
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“…In fact, since words which can be mapped to each other by automorphisms of F (in particular, conjugate words) represent the same permutations, we may assume that w is cyclically reduced. (See Proposition 3 of Silberger [12].) Also, the result is trivial if w has length 1, so assume that w has length greater than 1.…”
Section: Proof Of the Theorem Under Assumption ( * )mentioning
confidence: 99%
See 1 more Smart Citation
“…In fact, since words which can be mapped to each other by automorphisms of F (in particular, conjugate words) represent the same permutations, we may assume that w is cyclically reduced. (See Proposition 3 of Silberger [12].) Also, the result is trivial if w has length 1, so assume that w has length greater than 1.…”
Section: Proof Of the Theorem Under Assumption ( * )mentioning
confidence: 99%
“…D. Silberger [12] showed it for w = x p y q , p, q = 0, and asked whether it is true in general. M. Droste [3] showed it for w = w 1 w 2 , where w 1 and w 2 have no variables in common, and for w = x p y −r x q y r (and one can even require that the permutations used for x and y lie in certain specific conjugacy classes).…”
Section: Introductionmentioning
confidence: 99%
“…Lyndon [6] and Mycielski [9] recently settled a conjecture of Silberger [ 10] by showing that if G is the full symmetric group on an infinite set, and if w is not a power (greater than 1 ) of any element of F , then the equation w = g can always be solved in G. Adeleke and Holland [1] have recently proved results for the ordered analogue, i.e., when G is the automorphism group of a transitive chain. In this paper we consider the more general case when G is the order-automorphism group of a transitive tree, and we extend many of the results of Adeleke and Holland [1].…”
mentioning
confidence: 99%
“…In [10] Silberger asks if every w € Fx which is not a power, i.e., is not of the form vk with k > 1, can represent every tt in 5y, if F is infinite. He shows that this is true if w -xmyn for x, y 6 A, x ^ y and m ^ 0 ^ n. Another proof is given in [2].…”
mentioning
confidence: 99%
“…Ore), moreover, in this representation, x can be an involution. It is more tricky to represent the successor function in the form x2y2 (see [3]), and the theorem of Silberger [2,10] about xmyn is still harder. )…”
mentioning
confidence: 99%