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In article <1992Sep28.201204.13907@tamsun.tamu.edu> rlm7638@tamsun.tamu.edu (Jack McKinney) writes:
>     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)

There was a thread on sci.math where we talked about this.  There is
a nice construction due to Sylvester and Coxeter.  It has some cute
relationships to the icosahedron.  I will send you (and anyone else
who wants) what I have about this.

I am posting this for Allen Knutson, aknaton@math.Princeton.EDU,
who had trouble posting.

I'm curious about the differences between connectedness and path-connectedness.
If X is a connected topological space with two marked points, say that Y is
X-connected if for all y,y' \in Y one can map in X so as to have X's two
marked points hit y,y' respectively.

Now define a pre-partial order on connected spaces: "X connects more
than X'
does" if being X-connected implies being X'-connected. So for instance, all
R^n's (with whatever two marked points you like) are quickly seen to be
equivalent, but the topologist's sine curve connects strictly more than [0,1]
does.

This partial order (once we mod out by equivalence) plainly has a least
element: spaces where the two marked points coincide. Does it have any
maximal elements? Any canonical choice of representative from a given
connectedness class? Should one restrict to a smaller class of spaces
(say, compact Hausdorff) for some reason? How tall is it? (If people are
bugged by the category of connected spaces not being a set, I'm happy
to just take the ones of cardinality \leq 2^{\aleph_0} or whatever.)

References would be good too.                                   Allen K.

|> 
|>      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

According to the "Atlas of finite groups" by Conway et al.
 "The outer automorphism of S_6 interchanges the 15 duads ab (pairs)
  with the 15 synthemes ab.cd.ef (trisections), so interchanges the
  6 points a (which could be specified by 5 duads ab,ac,ad,ae,af)
  with the 6 totals (sets of 5 synthemes containing all duads)."

Richard Pinch, Dept of Pure Mathematics, University of Cambridge
               16 Mill Lane, Cambridge CB2 1SB, U.K.

In his article of 29-Sep-92 Jack McKinney writes:

         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)

S_6 operates on 6 points:

    1  2  3  4  5  6

Take the 15 duads (2-sets):

    12  13  14  15  16  23  24  25  26  34  35  36  45  46  56

Then there are 15 synthemes (3-sets of disjoint duads):

    12 34 56   13 24 56   14 23 56   15 23 46   16 23 45
    12 35 46   13 25 46   14 25 36   15 24 36   16 24 35
    12 36 45   13 26 45   14 26 35   15 26 34   16 25 34

Take the 6 pentads  (5-sets of synthemes which contain each duad exactely
once):

    12 34 56  13 25 46  14 26 35  15 24 36  16 23 45   (1')
    12 34 56  13 26 45  14 25 36  15 23 46  16 24 35   (2')
    12 35 46  13 24 56  14 25 36  15 26 34  16 23 45   (3')
    12 35 46  13 26 45  14 23 56  15 24 36  16 25 34   (4')
    12 36 45  13 24 56  14 26 35  15 23 46  16 25 34   (5')
    12 36 45  13 25 46  14 23 56  15 26 34  16 24 35   (6')

S_6  operates on those 6 pentads  and therefore on [1'..6'].  This  gives
the outer automorphism of S_6.

    (1,2)         -> (1',2')(3',4')(5',6')
    (1,2,3,4,5,6) -> (1')(2',3')(4',5',6')

There was a very nice article in 'sci.math' by John  C. Baez called "Some
thoughts on the number 6" about 4  months  ago,  from  which  most of the
above  is taken.  If you are  interested I  will see whether I can  still
find it and send it to you.

Martin.

--
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany

Apologies for the notational error in my posting. What I called synthemes are
totals, each made up of five synthemes.