Combinatorial categories and permutation groups

2013/09/24 by Gareth A. Jones, Jones, Gareth A.
Mathematics · #05A15 #05C10 #05E18 #20B25 (primary) #20J99 #57M10 (secondary) #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:05A15 #msc:05C10 #msc:05E18 #msc:20B25 #msc:20J99 #msc:57M10

paper · pdf · doi:10.48550/arxiv.1309.6119

arxiv created 2013/09/24 · arxiv updated 2013/09/25

Abstract

The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group Γ, with quotient group isomorphic to Γ/N. It is shown how to enumerate such objects with a given finite automorphism group G, how to represent them all as quotients of a single regular object U(G), and how they are acted on by the outer automorphism group of Γ. Examples constructed include kaleidoscopic maps with trinity symmetry.

Related

• Realisation of groups as automorphism groups in categories
• Telescopic groups and symmetries of combinatorial maps
• Automorphism groups of maps, hypermaps and dessins
• Enumerating Regular Objects associated with Suzuki Groups
• Structures induced on transversals to a subgroup of a group and hypergroups over the group