On representing words in the automorphism group of the random graph
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M Droste
und
J. K Truss
Abstract
We discuss the solubility of equations of the form w = g, where w is a word (an element of a free group FX) and g is an element of a given group G. A word for which this equation is soluble for every g ∈ G is said to be universal for G. It is conjectured that a word is universal for the automorphism group of the random graph if and only if it cannot be written as a proper power, corresponding to the results of [Randall Dougherty and Jan Mycielski. Representations of infinite permutations by words (II). Proc. Amer. Math. Soc.127 (1999), 2233–43.], [Roger C. Lyndon. Words and infinite permutations. In Mots, Lang. Raison Calc. (Hermès, 1990), pp. 143–152.], [Jan Mycielski. Representations of infinite permutations by words. Proc. Amer. Math. Soc. 100 (1987), 237–241.], where the same necessary and sufficient condition was established for infinite symmetric groups. We prove various special cases. A key ingredient is the use of ‘generic’ automorphisms, and elements which suitably approximate them, called ‘special’.
© Walter de Gruyter
Artikel in diesem Heft
- Crossover Morita equivalences for blocks of the covering groups of the symmetric and alternating groups
- On a theorem of Artin. II
- Cycle index methods for finite groups of orthogonal type in odd characteristic
- Characterization of injectors in finite soluble groups
- Some class size conditions implying solvability of finite groups
- On t-pure and almost pure exact sequences of LCA groups
- Translation equivalent elements in free groups
- On representing words in the automorphism group of the random graph
Artikel in diesem Heft
- Crossover Morita equivalences for blocks of the covering groups of the symmetric and alternating groups
- On a theorem of Artin. II
- Cycle index methods for finite groups of orthogonal type in odd characteristic
- Characterization of injectors in finite soluble groups
- Some class size conditions implying solvability of finite groups
- On t-pure and almost pure exact sequences of LCA groups
- Translation equivalent elements in free groups
- On representing words in the automorphism group of the random graph
