Direct product of automorphism groups of digraphs

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

Ars Mathematica Contemporanea, Tome 17 (2019) no. 1, pp. 89-101.

Résumé

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.

Détail

DOI : 10.26493/1855-3974.1498.77b

Keywords: Digraph, automorphism group, permutation group, direct product