Visualizing Automorphisms of S6
17 hours ago
- #permutations
- #group-theory
- #exceptional-automorphism
- Shuffling six items has a unique property called 'exceptional' or 'exotic' in mathematics.
- Permutations are rearrangements of items, and any arrangement can be achieved through swaps (transpositions).
- Cycle notation is used to describe permutations, where cycles like (1 2 3) represent a rotation of positions.
- Permutations form a group, which is a mathematical structure with associative operations, an identity element, and inverses.
- Groups can be isomorphic, meaning they have the same structure, and can be subgroups of larger groups.
- The symmetric group on n items (Sₙ) contains all permutations of n items and has subgroups of smaller symmetric groups.
- S₆ is exceptional because it has an automorphism (self-mapping preserving group structure) not just a relabeling, called an 'outer automorphism'.
- The exceptional automorphism of S₆ can be constructed by embedding S₅ into S₆ in a special way using 'pentads', 'synthemes', and 'duads'.
- A duad is a pair of items, a syntheme is a set of three duads covering all six items, and a pentad is a set of five synthemes covering all duads.
- Mystic pentagons provide a visual approach to understanding the exceptional automorphism, using complementary 5-cycles on a complete graph.