The extra outer automorphism in A 6 swaps the 3-cycles ( like ( 123 ) ) with elements of shape 3 2 ( like ( 123 ) ( 456 ) ).
32.
This is simply connected, has maximal compact subgroup the compact form ( see below ) of E 8, and has an outer automorphism group of order 2 generated by complex conjugation.
33.
The triality automorphism of Spin ( 8 ) lives in the outer automorphism group of Spin ( 8 ) which is isomorphic to the symmetric group S 3 that permutes these three representations.
34.
The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M12 as two different embeddings of M 11 in M 12, exchanged by an outer automorphism.
35.
The outer automorphism group of a finite simple group in some infinite family of finite simple groups can almost always be given by a uniform formula that works for all elements of the family.
36.
In one dimension, the only Lie algebra is the abelian Lie algebra "'R "'1, with outer automorphism group the group of non-zero real numbers.
37.
Conjugation by even elements are inner automorphisms of A " n " while the outer automorphism of A " n " of order 2 corresponds to conjugation by an odd element.
38.
The Schur multiplier of the Tits group is trivial and its outer automorphism group has order 2, with the full automorphism group being the group 2 " F " 4 ( 2 ).
39.
Thus Aut ( " S " 6 ) acts on these two conjugacy classes ( and the outer automorphism above interchanges these conjugacy classes ), and an index 2 subgroup stabilizes the transpositions.
40.
For " n " even there is instead an outer automorphism interchanging the two types of reflections ( properly, a class of outer automorphisms, which are all conjugate by an inner automorphism ).
