Outer automorphism group
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In mathematics, the outer automorphism group of a group G is the quotient Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete.
Note that the elements of Out(G) are cosets of automorphisms of G, and not themselves automorphisms. This is an instance of the fact that quotients of groups are not in general subgroups. In practice, however, elements of Aut(G) which are not inner automorphisms are often called outer automorphisms; they are representatives of the non-trivial cosets in Out(G).
The Schreier conjecture asserts that Out(G) is always a solvable group when G is a finite simple group. This result is now known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known.
This group is important in the topology of surfaces because there is a happy connection provided by the Dehn-Nielsen theorem: the extended mapping class group of the surface is the Out of its fundamental group.
Contents |
[edit] Out(G) for some finite groups
For the outer automorphism groups of all finite simple groups see the list of finite simple groups. Sporadic simple groups and alternating groups (other than the alternating group A6; see below) all have outer automorphism groups of order 1 or 2. The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for Dn(q) when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for D4(q) when it is the symmetric group on 3 points). These extensions are semidirect products except that for the Suzuki-Ree groups the graph automorphism squares to a generator of the field automorphisms.
| Group | Parameter | Out(G) | | Out(G) | |
|---|---|---|---|
| Z | infinite cyclic | Z2 | 2 |
| Zn | n > 2 | Zn× | φ(n) =  elements |
| Zpn | p prime, n > 1 | GLn(p) | (pn - 1)(pn - p )(pn - p2) ... (pn - pn-1)
elements |
| Sn | n not equal to 6 | trivial | 1 |
| S6 | Z2 (see below) | 2 | |
| An | n not equal to 6 | Z2 | 2 |
| A6 | Z2 × Z2(see below) | 4 | |
| PSL2(p) | p > 3 prime | Z2 | 2 |
| PSL2(2n) | n > 1 | Zn | n |
| PSL3(4) = M21 | Dih6 | 12 | |
| Mn | n = 11, 23, 24 | trivial | 1 |
| Mn | n = 12, 22 | Z2 | 2 |
| Con | n = 1, 2, 3 | trivial | 1 |
[edit] The outer 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[citation needed]: the alternating group A6 has outer automorphism group of order 4, rather than 2 for 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.
[edit] Outer automorphism groups of complex Lie groups
Let G now be a connected reductive group over an algebraically closed field. Then any two Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup. Associated to the Borel subgroup is a set of simple roots, and the outer automorphism may permute them, while preserving the structure of the associated Dynkin diagram. In this way one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G).
D4 has a very symmetric Dynkin diagram, which yields a large outer automorphism group of Spin(8), namely Out(Spin(8)) = S3; this is called triality.
[edit] Puns
The term "Outer automorphism" lends itself to puns: the term outermorphism is sometimes used for "outer automorphism", and a particular geometry on which 
acts is called outer space.
[edit] External links
(contains a lot of information on various classes of finite groups (in particular sporadic simple groups), including the order of Out(G) for each group listed.
