Outer Automorphism Group - The Outer Automorphisms of The Symmetric and Alternating Groups

The Outer Automorphisms of The Symmetric and Alternating Groups

For more details on this topic, see Automorphisms of the symmetric and alternating groups.

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. There is just one exception to this: the alternating group A6 has outer automorphism group of order 4, rather than 2 as do the other simple alternating groups (given by conjugation by an odd permutation). Equivalently the symmetric group S6 is the only symmetric group with a non-trivial outer automorphism group.

\begin{align} n\neq 6: \mathrm{Out}(S_n) & = 1 \\ n\geq 3,\ n\neq 6: \mathrm{Out}(A_n) & = C_2 \\ \mathrm{Out}(S_6) & = C_2 \\ \mathrm{Out}(A_6) & = C_2 \times C_2 \end{align}

Read more about this topic:  Outer Automorphism Group

Famous quotes containing the words outer and/or groups:

    “When human beings have been fascinated by the contemplation of their own hearts, the more intricate biological pattern of the female has become a model for the artist, the mystic, and the saint. When mankind turns instead to what can be done, altered, built, invented, in the outer world, all natural properties of men, animals, or metals become handicaps to be altered rather than clues to be followed.”
    —Margaret Mead (1901–1978)

    “screenwriter
    Policemen so cherish their status as keepers of the peace and protectors of the public that they have occasionally been known to beat to death those citizens or groups who question that status.”
    —David Mamet (b. 1947)