Here is, with Peter’s kind permission, a rebloging of Peter’s post on the automorphism group of . Other very nice accounts are by the Secret blogging seminar; John Baez,; A paper by Howard, Millson, Snowden, and Vakil; and most famously the legendary Chapter 6 (!) from the book by Cameron and Van-Lint (I dont have an electronic version for it).

My TYI 25 question about it arose from Sonia Balagopalan’s lecture in our combinatorics seminar on the 16 vertex triangulation of 4-dimensional projective space. (Here is the link to her paper.)

No account of the symmetric group can be complete without mentioning the remarkable fact that the symmetric group of degree *n* (finite or infinite) has an outer automorphism if and only if *n*=6.

Here are the definitions. An *automorphism* of a group *G* is a permutation *p* of the group which preserves products, that is, (*xy*)*p*=(*xp*)(*yp*) for all *x,y* (where, as a card-carrying algebraist, I write the function on the right of its argument). The automorphisms of *G* themselves form a group, and the *inner automorphisms* (the conjugation maps *x*?*g*^{-1}*xg*) form a normal subgroup; the factor group is the *outer automorphism group* of *G*. Abusing terminology, we say that *G* has outer automorphisms if the outer automorphism group is not the trivial group, that is, not all automorphisms are inner.

Now the symmetric group *S*

View original post 1,245 more words

Yet another very nice account is given in this post by Richard Green “The exceptional symmetry” https://plus.google.com/u/0/101584889282878921052/posts/ioQW2zGjwwM

Pingback: New Year’s Greeting from Franz Kafka « Log24