Identical relations in symmetric groups and separating words with reversible automata

Authors:
R. A. Gimadeev;M. N. Vyalyi
Affiliations:
Moscow Institute of Physics and Technology;Dorodnitsyn Computing Center of RAS
Venue:
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
Year:
2010

Citing 4
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Abstract

Separating words with automata is a longstanding open problem in combinatorics on words. In this paper we present a related algebraic problem. What is the minimal length of a nontrivial identical relation in the symmetric group Sn? Our main contribution is an upper bound $2^{O(\sqrt n\log n)}$ on the length of the shortest nontrivial identical relation in Sn. We also give lower bounds for words of a special types. These bounds can be applied to the problem of separating words by reversible automata. In this way we obtain an another proof of the Robson’s square root bound.