Automorphisms of the symmetric and alternating groups
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.
Among symmetric groups, only S6 has a (non-trivial) outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S6) = C2.
This was discovered by Otto Hölder in 1895.
This also yields another outer automorphism of A6, and this is the only exceptional outer automorphism of a finite simple group: for the infinite families of simple groups, there are formulas for the number of outer automorphisms, and the simple group of order 360, thought of as A6, would be expected to have two outer automorphisms, not four. However, when A6 is viewed as PSL(2, 9) the outer automorphism group has the expected order. (For sporadic groups (not falling in an infinite family), the notion of exceptional outer automorphism is ill-defined, as there is no general formula.)
There are numerous constructions, listed in (Janusz & Rotman 1982).
Note that as an outer automorphism, it's a class of automorphisms, well-determined only up to an inner automorphism, hence there is not a natural one to write down.
One method is:
Throughout the following, one can work with the multiplication action on cosets or the conjugation action on conjugates.
To see that S6 has an outer automorphism, recall that homomorphisms from a group G to a symmetric group Sn are essentially the same as actions of G on a set of n elements, and the subgroup fixing a point is then a subgroup of index at most n in G. Conversely if we have a subgroup of index n in G, the action on the cosets gives a transitive action of G on n points, and therefore a homomorphism to Sn.
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