Exceptional Object - Outer Automorphisms

Outer Automorphisms

Certain families of groups generically have a certain outer automorphism group, but in particular cases they have other, exceptional outer automorphisms.

Among families of finite simple groups, the only example is in the automorphisms of the symmetric and alternating groups: for the alternating group has one outer automorphism (corresponding to conjugation by an odd element of ) and the symmetric group has no outer automorphisms. However, for there is an exceptional outer automorphism of (of order 2), and correspondingly, the outer automorphism group of is not (the group of order 2) but rather (the Klein four-group).

If one instead considers A₆ as the (isomorphic) projective special linear group PSL(2,9), then the outer automorphism is not exceptional; thus the exceptionalness can be seen as due to the exceptional isomorphism This exceptional outer automorphism is realized inside of the Mathieu group M₁₂ and similarly, M₁₂ acts on a set of 12 elements in 2 different ways.

Among Lie groups, the spin group Spin(8) has an exceptionally large outer automorphism group (namely ), which corresponds to the exceptional symmetries of the Dynkin diagram D₄. This phenomenon is referred to as triality.

The exceptional symmetry of the D₄ diagram also gives rise to the Steinberg groups.