Groups

Some pages on groups.

• Cayley Diagrams
  • Small Groups giving Cayley diagrams for all groups with fewer than 32 elements:
    • on the plane with the arcs crossing as necessary;
    • SVG version of above which looks better, but renders more slowly, and does not work with older browsers
    • on the torus;
    • SVG version of above which looks better, but renders more slowly, and does not work with older browsers
    • on the plane but excluding direct products, and using orange triangles to denote central extensions.
  • Platonic Solids giving Cayley diagrams drawn on symmetrical polyhedra
  • Plane Tilings giving Cayley diagrams drawn on the Euclidean plane
  • Infinite "Two-Ended" Groups giving Cayley diagrams for a very few infinite groups
  • Double Cayley diagrams, showing groups and their automorphism groups in the same diagram:
    • Small groups
    • Small groups of prime order
  • Cayley diagrams listed by the genus of the surface they are presented on:
    • the sphere,
    • the torus,
    • the genus-2 surface,
    • the genus-3 surface.
  • the Pauli group
  • PSL(2,7) ≅ GL(3,2)
  • Some thoughts about Cayley diagrams
• group extensions, with
  • Examples showing all 14 ways to build a group of order 8
• ways of building larger groups from small ones, with sub-pages
  • Semi-direct products
  • Toll-bean Extensions introducing the symbols and ↑
  • Valid and Invalid Cayley Diagrams
• some groups acting on the colour cube: S4, S4 × C2, SO(3).
• miscellaneous short pages
• Puzzles
  • Puzzle of the Day
  • A puzzle involving permutations

Here are a few links to pages about groups.

Regular Maps

Index page for regular maps in surfaces of orientable genera 0, 1, 2, 3, 4; and non-orientable genera 1, 4, 5, 6.

Some old pages on regular maps on surfaces of various genus:

• sphere
• torus
• genus-2 surface
• genus-3 surface