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dc.contributor.authorAraújo, João
dc.contributor.authorCameron, Peter Jephson
dc.contributor.authorSteinberg, Benjamin
dc.identifier.citationAraújo , J , Cameron , P J & Steinberg , B 2017 , ' Between primitive and 2-transitive : synchronization and its friends ' , EMS Surveys in Mathematical Sciences , vol. 4 , no. 2 , pp. 101-184 .
dc.identifier.otherPURE: 250259398
dc.identifier.otherPURE UUID: 846e6e20-402f-481a-b4d9-8c526cc840f0
dc.identifier.otherORCID: /0000-0003-3130-9505/work/58055739
dc.identifier.otherWOS: 000417988900001
dc.descriptionThe second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013en
dc.description.abstractAn automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.
dc.relation.ispartofEMS Surveys in Mathematical Sciencesen
dc.rights© 2017, European Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at
dc.subjectPrimitive groupsen
dc.subjectSynchronizing groupsen
dc.subjectSpreading groupsen
dc.subjectSeparating groupsen
dc.subjectČerný conjectureen
dc.subjectOvoids and spreadsen
dc.subjectTransformation semigroupsen
dc.subjectQA Mathematicsen
dc.titleBetween primitive and 2-transitive : synchronization and its friendsen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews. Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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