On primitive 2-closed permutation groups of rank at most four

2021/10/15 by Michael Giudici, Luke Morgan, Giudici, Michael +4
Computer Science · Engineering · Mathematics · #05E18 #20B25 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems #math.CO #math.GR #msc:05E18 #msc:20B25

paper · pdf · doi:10.48550/arxiv.2110.07896

openalex publication_date 2021/10/15 · arxiv created 2022/09/17 · arxiv updated 2022/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We characterise the primitive 2-closed groups G of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or G\leqslant AΓL₁(p^d), the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.

Citations

• The Primitive Permutation Groups of Degree Less Than 4096
• Finite Permutation Groups and Finite Simple Groups

Related

• New 2-closed groups that are not automorphism groups of digraphs
• THE PRIMITIVE PERMUTATION GROUPS OF SQUAREFREE DEGREE
• The Primitive Permutation Groups of Degree Less Than 4096
• Relatively closed subgroups of permutation groups with a cyclic regular normal subgroup
• The automorphism groups of small affine rank 3 graphs