🚀 go-pugleaf

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