Direct product of automorphism groups of digraphs

Mariusz Grech, Wilfried Imrich, Anna Dorota Krystek, Łukasz Jan Wojakowski

Abstract


We study the direct product of automorphism groups of digraphs, where automorphism groups are considered as permutation groups acting on the sets of vertices. By a direct product of permutation groups (A, V) × (B, W) we mean the group (A × B, V × W) acting on the Cartesian product of the respective sets of vertices. We show that, except for the infinite family of permutation groups Sn × Sn, n ≥ 2, and four other permutation groups, namely D4 × S2, D4 × D4, S4 × S2 × S2, and C3 × C3, the direct product of automorphism groups of two digraphs is itself the automorphism group of a digraph. In the course of the proof, for each set of conditions on the groups A and B that we consider, we indicate or build a specific digraph product that, when applied to the digraphs representing A and B, yields a digraph whose automorphism group is the direct product of A and B.


Keywords


Digraph, automorphism group, permutation group, direct product

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DOI: https://doi.org/10.26493/1855-3974.1498.77b

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications