Direct product of automorphism groups of digraphs - Grech, Mariusz (Ars mathematica contemporanea, 2019)
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 ?$S_n × S_n$?, ?$n \ge 2$?, and four other permutation groups, namely ?$D_4 × S_2$?, ?$D_4 × D_4$?, ?$S_4 × S_2 × S_2$?, and ?$C_3 × C_3$?, 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: automorphism group; digraf; digraph; direct product; direktni produkt; grupa avtomorfizmov; permutacijska grupa; permutation group