Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks
Jo\~ao Ara\'ujo, Peter J. Cameron

TL;DR
This paper advances the understanding of primitive groups' ability to synchronize non-uniform maps, especially those with extreme ranks, by extending known results and providing new characterizations through graph-theoretic methods.
Contribution
It proves that primitive groups synchronize maps with kernel type (k,1,...,1) and maps of ranks 3, 4, and n-2, extending previous results and offering new insights into group synchronization.
Findings
Primitive groups synchronize maps with kernel type (k,1,...,1).
Primitive groups synchronize maps of ranks 3, 4, and n-2.
Graph-theoretic techniques are used to analyze synchronization properties.
Abstract
Let be a set of cardinality , a permutation group on , and a map which is not a permutation. We say that synchronizes if the semigroup contains a constant map. The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree primitive groups synchronize maps of rank (thus, maps with kernel type ). We prove some extensions of Rystsov's result, including this: a primitive group synchronizes every map whose kernel type is . Incidentally this result provides a new characterization of imprimitive groups. We also prove that the conjecture above holds for maps of extreme ranks, that is, ranks 3, 4 and . These proofs use a graph-theoretic technique due to the second…
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Taxonomy
Topicssemigroups and automata theory · Geometric and Algebraic Topology · Finite Group Theory Research
