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 An, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering An as a subgroup of the symmetric group Sn conjugation by any odd permutation is an outer automorphism of An or more precisely "represents the class of the (non-trivial) outer automorphism of An", 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.
- 1 Out(G) for some finite groups
- 2 The outer automorphisms of the symmetric and alternating groups
- 3 Outer automorphism groups of reductive algebraic groups
- 4 Outer automorphism groups of complex and real simple Lie algebras
- 5 Structure
- 6 Dual to center
- 7 Applications
- 8 Puns
- 9 External links
- 10 See also
- 11 References
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 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.
|Z||infinite cyclic||Z2||2; the identity and the map f(x) = -x|
|Zn||n > 2||Zn×||φ(n) = elements; one corresponding to multiplication by an invertible element in Zn viewed as a ring.|
|Zpn||p prime, n > 1||GLn(p)||(pn − 1)(pn − p )(pn − p2) ... (pn − pn−1)
|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(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|
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: 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.
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
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).
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 , the automorphism group is a semidirect product of and , i.e., the short exact sequence
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 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:
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.
(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.
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