on Out(Sym₆)

The existence, unique among all finite symmetry groups, of the outer automorphism on Sym₆ is one of those remarkable and relatively straightforward results I am always surprised to learn is not widespread among people much more technically learned and adept than myself. While not trivial, the proofs are quite quick, especially by comparison with the sprawling demonstrations sometimes found elsewhere on exceptional objects in the theory of finite groups. (The standard proofs do, however, at least for me have a bit of the feeling of a magic trick: a reflection, probably, of the exceptional character of the object being constructed.)

Here are a few resources I found helpful:

• The Wikipedia page gives a handy rundown, including a particularly accessible presentation of the Sylvester-style graph construction used in this game, but is frustratingly vague on the significance of a transitive Sym₅-isomorphic subgroup of Sym₆.
• This short and snappy article by Janusz and Rotman (available wherever fine academic texts are pirated) helps explain just that, as well as giving some more examples of the highly varied proofs and constructions found to date of the automorphism (a testament to the low numbers involved and the ubiquity of the symmetry groups). There is a bunch of subsequent expository work citing it that also looks interesting. It assumes a bit of background.
• John Baez has a very fun and accessible (albeit dense) page up dealing with the automorphism, managing in the first and most accessible section to fuse the graph theoretic and monomorphism-of-S₅ approaches by way of an interesting geometric argument using the Platonic icosahedron.
• I have not yet read through this James Sylvester paper (hosted by Baez), but it looks to include his original and independent discovery of the automorphism. All in a delightfully eccentric, 19th century English style. (While "duad", "syntheme", and "synthematic total" never caught on, they have a similar vibe to some of his more successful neologisms: "graph", "matrix", "totient", "discriminant". Good stuff!)

One point I have found worth emphasising on a first encounter is that the outer (that is, non-inner; that is, not arising from conjugation) automorphism on Sym₆ is unique only in that the order of the quotient group Aut(Sym₆)/Inn(Sym₆) is 2 (the other element being the quotient group identity, the subgroup Inn(Sym₆) itself). The outer automorphism is thus unique only up to an inner automorphism. Nor can you designate a privileged outer automorphism corresponding to the identity inner automorphism, for there is no canonical pairing between the inner and outer automorphisms; rather, any outer automorphism can be taken to any other by some conjugation or other. I have emphasised this fact by randomising the particular automorphism in any given session of the game.

Truly the automorphism is a thing of beauty! Perhaps it is what so especially pleased God to behold once he had wrought six days to permute.

Back to the game!