The Exceptional Symmetry

This note gives an elementary proof that the symmetric groups possess only one exceptional symmetry. I am referring to the fact that the outer automorphism group of the symmetric group Symn is trivial unless n = 6 and the outer automorphism group of Sym6 has a unique nontrivial element. When we study symmetric groups, we often invoke their natural faithful representation as permutations of a set without a second thought, but to what extent is this representation intrinsic to the structure of the group and to what extend is it one of several possible choices available? Concretely, suppose I am studying the permutations SymX of a set X = {1, 2, 3, 4, 5, 6} and you are studying the permutations SymA of a set A = {a, b, c, d, e, f} and suppose further that we know an explicit isomophism φ between my group SymX and your group SymA. Does this means that there is a way to identify my set X with your set A which gives rise to the isomorphism φ? In other words, must my transpositions correspond to your transpositions? Must my 3-cycles correspond to your 3-cycles? Or might it be possible that the transposition (1, 2) in my group is sent by the isomorphism φ to the element (a, b)(c, d)(e, f) in your group? The goal of this note is to give an elementary proof of the fact that yes there is an isomorphism φ between these two specific groups sending (1, 2) to (a, b)(c, d)(e, f), but that this is essentially the only unexpected isomorphism among all of the symmetric groups. In the language of outer automorphism groups (which we recall below) we give a proof of the following well-known and remarkable fact. Theorem. Out(Symn) is trivial for n 6= 6 and Z/2Z when n = 6. Recall that the set of all isomorphisms from a group G to itself form a group Aut(G) under composition called its automorphism group. Moreover, in any group we can create an automorphism by conjugating by a fixed element of G. Such automorphisms are called inner automorphisms and they form a subgroup Inn(G) which is normal in Aut(G). These are what one might call the “expected” automorphisms. Note that in the case of the symmetric groups, conjugating by a permutation corresponds to relabeling the elements of the set on which it acts. The quotient group Out(G) := Aut(G)/Inn(G) is the group of outer automorphisms. When Date: December 4, 2014.