Outer automorphism group

The Schreier conjecture asserts that {{math|Out(G)}} is always a solvable group when {{mvar|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.

The outer automorphism group is dual to the center in the following sense: conjugation by an element of {{mvar|G}} is an automorphism, yielding a map {{math|σ : G → 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 exact sequence

$$Z(G)\; \backslash hookrightarrow\; G\; \backslash ,\; \backslash overset\{\backslash sigma\}\{\backslash longrightarrow\}\; \backslash ,\; \backslash mathrm\{Aut\}(G)\; \backslash twoheadrightarrow\; \backslash mathrm\{Out\}(G)$$

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

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 outer automorphism group of its fundamental group.

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, {{math|A{{sub|6}}}}; 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 {{math|list of finite simple groups#Dn.28q.29 n .3E 3 Chevalley groups.2C orthogonal groups}}, 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 {{math|D{{sub|4}}(q)}}, when it is the symmetric group on 3 points). These extensions are not always semidirect products, as the case of the alternating group {{math|A{{sub|6}}}} shows; a precise criterion for this to happen was given in 2003.A. Lucchini, F. Menegazzo, M. Morigi (2003), "[https://projecteuclid.org/download/pdf_1/euclid.ijm/1258488162 On the existence of a complement for a finite simple group in its automorphism group]", Illinois J. Math. 47, 395–418.

{{details|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:ATLAS p. xvi the alternating group {{math|A{{sub|6}}}} 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 {{math|S{{sub|6}}}} is the only symmetric group with a non-trivial outer automorphism group.

: $\backslash begin\{align\}$

n \neq 6: \operatorname{Out}(\mathrm{S}_n) & = \mathrm{C}_1 \\

n \geq 3,\ n \neq 6: \operatorname{Out}(\mathrm{A}_n) & = \mathrm{C}_2 \\

\operatorname{Out}(\mathrm{S}_6) & = \mathrm{C}_2 \\

\operatorname{Out}(\mathrm{A}_6) & = \mathrm{C}_2 \times \mathrm{C}_2

\end{align}

Note that, in the case of {{math|G {{=}} A{{sub|6}} {{=}} PSL(2, 9)}}, the sequence {{math|1 ⟶ G ⟶ Aut(G) ⟶ Out(G) ⟶ 1}} does not split. A similar result holds for any {{math|PSL(2, q{{sup|2}})}}, {{mvar|q}} odd.

File:Dynkin diagram D4.png, {{math|D{{sub|4}}}}, correspond to the outer automorphisms of {{math|Spin(8)}} in triality.]]

Let {{mvar|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 {{mvar|G}} with a subgroup of {{math|Out(G)}}.

{{math|D{{sub|4}}}} has a very symmetric Dynkin diagram, which yields a large outer automorphism group of {{math|Spin(8)}}, namely {{math|Out(Spin(8)) {{=}} S{{sub|3}}}}; this is called triality.

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, {{mvar|𝔤}}, the automorphism group {{math|Aut(𝔤)}} is a semidirect product of {{math|Inn(𝔤)}} and {{math|Out(𝔤)}}; i.e., the short exact sequence

: {{math|1 ⟶ Inn(𝔤) ⟶ Aut(𝔤) ⟶ Out(𝔤) ⟶ 1}}

splits. In the complex simple case, this is a classical result,{{citation| last1 = Fulton

| first1 = William

| author1-link = William Fulton (mathematician)

| last2 = Harris

| first2 = Joe

| author2-link = Joe Harris (mathematician)

| year = 1991

| title = Representation theory. A first course

| publisher = Springer-Verlag

| location = New York

| series = Graduate Texts in Mathematics, Readings in Mathematics

| volume = 129

| isbn = 978-0-387-97495-8

| doi = 10.1007/978-1-4612-0979-9

| oclc = 246650103

| language = en-gb

| mr = 1153249|contribution=Proposition D.40}} whereas for real simple Lie algebras, this fact was proven as recently as 2010.[http://www.heldermann.de/JLT/JLT20/JLT204/jlt20035.htm JLT20035]

The term outer automorphism lends itself to word play: the term outermorphism is sometimes used for outer automorphism, and a particular geometry on which {{math|Out(F{{sub|n}})}} acts is called outer space.

• Mapping class group
• Out(Fn)
• Outer space

{{More citations needed|date=November 2009}}

{{reflist}}

• [http://brauer.maths.qmul.ac.uk/Atlas/v3/ 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 {{math|Out(G)}} for each group listed.

Category:Group theory

Category:Group automorphisms