Outer automorphism group

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.

An automorphism of a group which is not inner is called an outer automorphism. 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 (isomorphic to) subgroups. Elements of Out(G) are cosets of Inn(G) in Aut(G).

For example, for the alternating group A_n, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering A_n as a subgroup of the symmetric group S_n conjugation by any odd permutation is an outer automorphism of A_n or more precisely "represents the class of the (non-trivial) outer automorphism of A_n", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.

However, for an abelian group A, the inner automorphism group is trivial and thus the automorphism group and outer automorphism group are naturally identified, and outer automorphisms do act on A.

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 D_n(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 not always semidirect products, as the case of the alternating group A6 shows; a precise criterion for this to happen is given in : A. Lucchini, F. Menegazzo and M. Morigi, On the existence of a complement for a finite simple group in its automorphism group, Illinois J. Math. 47 (2003), 395-418.

Group	Parameter	Out(G)	$|\mbox{Out}(G)|$
Z	infinite cyclic	Z2	2; the identity and the map f(x) = -x
Zn	n > 2	Zn^×	φ(n) = $n\prod_{p|n}\left(1-\frac{1}{p}\right)$ elements; one corresponding to multiplication by an invertible element in Zn viewed as a ring.
Zp^n	p prime, n > 1	GLn(p)	(p^n − 1)(p^n − p )(p^n − p^2) ... (p^n − p^n−1) elements
Sn	n ≠ 6	trivial	1
S6		Z2 (see below)	2
An	n ≠ 6	Z2	2
A6		Z2 × Z2(see below)	4
PSL2(p)	p > 3 prime	Z2	2
PSL2(2^n)	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

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:^[1] 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.

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

Note that, in the case of G = A₆ = PSL(2,9), the sequence 1 -> G -> Aut(G) -> Out(G) -> 1 does not split. A similar result holds for any PSL(2,q^2), q odd.

Outer automorphism groups of reductive algebraic groups

The symmetries of the Dynkin diagram D₄ correspond to the outer automorphisms of Spin(8) in triality.

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.

Outer automorphism groups of complex and real simple Lie algebras

The preceding interpretation of outer automorphisms as symmetries of a Dynkin diagram follows from the general fact, that for a complex or real simple Lie algebra $\mathfrak{g}$, the automorphism group $\operatorname{Aut}(\mathfrak{g})$ is a semidirect product of $\operatorname{Inn}(\mathfrak{g})$ and $\operatorname{Out}(\mathfrak{g})$, i.e., the short exact sequence

$1 \;\xrightarrow{}\; \operatorname{Inn}(\mathfrak{g}) \;\xrightarrow{}\; \operatorname{Aut}(\mathfrak{g}) \;\xrightarrow{}\; \operatorname{Out}(\mathfrak{g}) \;\xrightarrow{}\; 1$

splits. In the complex simple case, this is a classical result,^[2] whereas for real simple Lie algebras, this fact has been proven as recently as 2010.^[3]

Structure

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.

Dual to center

The outer automorphism group is dual to the center in the following sense: conjugation by an element of G is an automorphism, yielding a map $\sigma\colon G \to \operatorname{Aut}(G).$ The kernel of the conjugation map is the center, while the cokernel is the outer automorphism group (and the image is the inner automorphism group). This can be summarized by the short exact sequence:

$Z(G) \hookrightarrow G \overset{\sigma}{\to} \operatorname{Aut}(G) \twoheadrightarrow \operatorname{Out}(G).$

Applications

The outer automorphism group of a group acts on conjugacy classes, and accordingly on the character table. See details at character table: outer automorphisms.

Topology of surfaces

The outer automorphism group is important in the topology of surfaces because there is a connection provided by the Dehn–Nielsen theorem: the extended mapping class group of the surface is the Out of its fundamental group.

Puns

The term "outer automorphism" lends itself to puns: the term outermorphism is sometimes used for "outer automorphism", and a particular geometry on which $\scriptstyle\operatorname{Out}(F_n)$ acts is called outer space.

External links

• ATLAS of Finite Group Representations-V3

(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.

See also

• Mapping class group
• Out(F_n)

References

1. ATLAS p. xvi
2. (Fulton & Harris 1991, Proposition D.40)