TY - JOUR

T1 - Direct product of automorphism groups of digraphs

AU - Grech, Mariusz

AU - Imrich, Wilfried

AU - Krystek, Anna Dorota

AU - Wojakowski, Łukasz Jan

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Automorphism group

KW - Digraph

KW - Direct product

KW - Permutation group

UR - http://www.scopus.com/inward/record.url?scp=85068332720&partnerID=8YFLogxK

U2 - 10.26493/1855-3974.1498.77b

DO - 10.26493/1855-3974.1498.77b

M3 - Article

AN - SCOPUS:85068332720

VL - 17.2019

SP - 89

EP - 101

JO - Ars mathematica contemporanea

JF - Ars mathematica contemporanea

SN - 1855-3966

IS - 1

ER -