Combinatorial categories and permutation groups. (English) Zbl 1395.20002
Summary: 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 \(\Gamma\), with automorphism group isomorphic to \(\Gamma / N\). 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 \(\mathcal{U}(G)\), and how the outer automorphism group of \(\Gamma\) acts on them. Examples constructed include kaleidoscopic maps with trinity symmetry.
MSC:
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
| 05A15 | Exact enumeration problems, generating functions |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 05E18 | Group actions on combinatorial structures |
| 18B99 | Special categories |
| 20J99 | Connections of group theory with homological algebra and category theory |
| 57M10 | Covering spaces and low-dimensional topology |
