🚀 go-pugleaf

|> 
|>      A complete group is one that is isomorphic to its group of
|> Automorphisms. The symmetric group on n elements is complete for
|> all integers n _except_ 2 and 6. It is also true that the outer
|> automorphisms of S_6 form one coset to the set of inner automorphisms.
|> Can someone provide an outer automorphism on S_6 ? (This would
|> completely specify Aut(S_6), which is isomorphic to S_6 x Z_2)

No it isn't. It is of order 240 and contains a normal subgroup isomorphic to S_6, but it is not isomorphic to S_6 x Z_2.

I can't remember exactly how the outer automorphism acts, but you can reconstruct
it by observing that S_6 contains just six synthemes. A syntheme is a set of five elements of the form (a,b)(c,d)(e,f) which together contain all fifteen possible pairs eg:

(1,2)(3,4)(5,6)  (1,3)(2,5)(4,6) (1,4)(2,6)(3,5) (1,5)(2,4)(3,6) and
(1,6)(2,3)(4,5)

Any element of S_6 permutes these six synthemes by conjugation and the permutation it performs can be taken to be its image under the automorphism.

		
			Steve Linton