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:

    “I know, it must have been my imagination, but it makes me realize how desperately alone the Earth is. Hanging in space like a speck of food floating in the ocean. Sooner or later to be swallowed up by some creature floating by.... Time will tell, Dr. Mason. We can only wait and wonder. Wonder how, wonder when.”
    —Tom Graeff. Young astronomer, Teenagers from Outer Space, after just seeing the invading spaceship through his telescope, and dismissing it (1959)

    “In America every woman has her set of girl-friends; some are cousins, the rest are gained at school. These form a permanent committee who sit on each other’s affairs, who “come out” together, marry and divorce together, and who end as those groups of bustling, heartless well-informed club-women who govern society. Against them the Couple of Ehepaar is helpless and Man in their eyes but a biological interlude.”
    —Cyril Connolly (1903–1974)