Finite digraphs with given regular automorphism groups
Copyright policy )Finite digraphs with given regular automorphism groups
We prove that, given an infinite group G there is a directed graph X such that its automorphism group A(X) is the regular representation of G. The proof uses combinatorial set theoretic arguments (Erdös-Rado partition calculus). One of the lemmas asserts that, if |G|>2κ then G contains a subset H of power κ+ satisfying x−1y=y−1z⇒x=z (x, y, z ∈ H) Corollary. Given an infinite group G there exist a simple graph X a partially ordered set P, and a commutative semigroup S such that A(X)≅A(P)≅A(S)≅G. Moreover A(X) and A(P) each have three orbits, while A(S) has five orbits.
- Eötvös Loránd University Hungary
regular permutation group, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Digraph, Regular Representation, Directed graphs (digraphs), tournaments, automorphism group, Given Regular Automorphism Groups, Graph, digraph, Graphs and abstract algebra (groups, rings, fields, etc.), Theoretical Computer Science, Distributive lattices, Erdoes-Rado Partition Calculus, Partial orders, general, Computational Theory and Mathematics, digraphical regular representation, Infinite Digraphs, Infinite Group, Hypergraph, Discrete Mathematics and Combinatorics, Semigroups, DRR
regular permutation group, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Digraph, Regular Representation, Directed graphs (digraphs), tournaments, automorphism group, Given Regular Automorphism Groups, Graph, digraph, Graphs and abstract algebra (groups, rings, fields, etc.), Theoretical Computer Science, Distributive lattices, Erdoes-Rado Partition Calculus, Partial orders, general, Computational Theory and Mathematics, digraphical regular representation, Infinite Digraphs, Infinite Group, Hypergraph, Discrete Mathematics and Combinatorics, Semigroups, DRR
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